category: literaturenote citekey: bootdiffusionindexforecastsunder2025 title: "Diffusion index forecasts under weaker loadings: PCA, ridge regression, and random projections" authors: "Boot, Tom; Keijsers, Bart" year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.09575v1" zotero_key: EEPN4486 zotero_storage: HFQ8A4AV collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Boot, Tom"

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Diffusion index forecasts under weaker loadings: PCA, ridge regression, and random projections

Tom Boot University of Groningen t.boot@rug.nl

Bart Keijsers University of Amsterdam b.j.l.keijsers@uva.nl

June 12, 2025

Abstract

We study the accuracy of forecasts in the diffusion index forecast model with possibly weak loadings. The default option to construct forecasts is to estimate the factors through principal component analysis (PCA) on the available predictor matrix, and use the estimated factors to forecast the outcome variable. Alternatively, we can directly relate the outcome variable to the predictors through either ridge regression or random projections. We establish that forecasts based on PCA, ridge regression and random projections are consistent for the conditional mean under the same assumptions on the strength of the loadings. However, under weaker loadings the convergence rate is lower for ridge and random projections if the time dimension is small relative to the cross-section dimension. We assess the relevance of these findings in an empirical setting by comparing relative forecast accuracy for monthly macroeconomic and financial variables using different window sizes. The findings support the theoretical results, and at the same time show that regularization-based procedures may be more robust in settings not covered by the developed theory.

Keywords: diffusion index forecasting, principal component analysis, ridge regression, random projections.

JEL codes: C22, C38, C53.

1 Introduction

The diffusion index forecasting model developed by Stock and Watson (1998, 2002a) provides a parsimonious approach to use information from large macroeconomic data sets such as the monthly FRED-MD and quarterly FRED-QD (McCracken and Ng, 2016, 2020). To establish consistency of diffusion index forecasts for the conditional mean, the classical assumption is that the factors are 'pervasive' in the sense that these factors drive most variation in the available predictors. When there are r factors, this pervasiveness is reflected in a sharp drop in the eigenvalues of the predictors' covariance matrix after the rth largest eigenvalue. In the absence of such a drop, PCA fails to consistently estimate the factors (Onatski, 2012), and hence, yields inconsistent forecasts.

Recently, Bai and Ng (2023) provide an analysis of PCA-based factor estimation in the intermediate regime where the drop in the eigenvalues is 'small'. In this paper, we utilize their results to analyze the consistency of diffusion index forecasts across three different estimation methods. First, we examine forecasts based on factors estimated by PCA. When the factor loadings are strong, the results naturally coincide with those in Stock and Watson (2002a). Under weaker loadings, we show that the convergence rate depends on the loadings' strength as well as on the relative growth of the cross-sectional dimension (N) and the time dimension (T). The convergence rate tends to drop when N increases faster than T. We also show that a substantial improvement in the convergence rate can be obtained when the idiosyncratic errors in the factor model are temporally independent.

Next, we explore two alternative regularization-based forecast methods: ridge regression forecasts and random projection based forecasts. The former were analyzed under a factor model structure by De Mol et al. (2008) as a form of Bayesian shrinkage. The latter can also be expected to work well when the data exhibits a factor structure (Boot and Nibbering, 2019). We provide a unified analysis of both methods by deriving bounds on the implicit regularization of the forecast induced by the choice of the ridge penalty and the random projection dimension. Both methods effectively soft-threshold the eigenvalues of the predictor matrix, unlike PCA-based methods that apply hard-thresholding.

We find that appropriate selection of regularization parameters ensures the consistency of both regularization-based forecasts for the conditional mean under the same conditions as the PCA-based forecast. Under strong loadings, the results improve the convergence rates for ridge regression forecasts established by De Mol et al. (2008). We also show that the convergence rate can be negatively impacted by regularization bias under weaker loadings, particularly when the number of predictors grows faster than the sample size. In this case, the convergence rate can be slower relative to PCA.

We provide two sets of simulations. The crucial difference between these simulations is the presence of serial correlation in the idiosyncratic errors. Without serial correlation, the simulations are in line with the asymptotic results even for small samples. If there is autocorrelation in the idiosyncratic errors, then for weak loadings and small sample sizes, the accuracy of PCA seems to falter, while that of the regularization methods stays remarkably stable. Part of the deterioration of PCA can be explained by the fact that in small samples the nominal number of factors may be lower than the optimal number of factors. When we increase the dimensions of the data, the accuracy of PCA again improves over ridge regression and random projections.

The simulations tie in with our empirical application on forecasting monthly and quarterly macroeconomic and financial series from the FRED-MD and FRED-QD (McCracken and Ng, 2016, 2020). We estimate a diffusion index forecast model and manipulate the ratio of the sample size of the number of predictors in two ways while keeping the number of variables fixed: by choosing different estimation windows and by comparing across frequencies. In line with the developed theory, we find that as we decrease the moving window size, PCA forecasts become relatively more accurate compared to the regularization methods. At the same time, as in Boot and Nibbering (2019), and our simulations results for small samples with serial correlation in the idiosyncratic errors, random projections and ridge frequently outperform the PCA-based forecast. PCA yields better forecasts in about 10 percent of the cases when T > N, while PCA-based forecasts are more accurate than ridge and random projections for close to 40 percent of the variables when T = ³N. In addition, in the monthly dataset, PCA-based forecasts are more accurate in 7 to 13 percent of the cases, but this increases to 19 percent when using quarterly data.

Related literature The possible presence of weak loadings, for example in macroeconomic and financial applications, has triggered an active literature. Onatski (2012) considers an asymptotic regime where the eigenvalues are not well-separated, as in the spiked covariance matrix model of Johnstone and Lu (2009) and Paul (2007). There, the ratio of the leading eigenvalues of the covariance matrix of the predictors with the smaller eigenvalues is assumed to be bounded. This constitutes an even weaker setting relative to the one considered in Bai and Ng (2023) that we follow in this paper.

Weak loadings as defined in the current paper can arise through a sparse structure of the loading matrix. Estimation and inference in this setting is considered in Uematsu and Yamagata (2022a) and Uematsu and Yamagata (2022b). Fan and Liao (2022) consider a setting with weak loadings that are not necessarily sparse. The approach is somewhat similar to the random projection forecast as they use a cross-sectional projection to estimate the factors. The results presented there require the weights of the projections to be bounded (Assumption 2.1.i in Fan and Liao (2022)). Also, instead of using a single projection, we consider averaging over multiple random projections to eliminate additional variance from the fact that we project onto a random subspace. Karabiyik and Westerlund (2021) consider simple averages of predictors within a pre-specified group to recover group-specific factors. In the context of the FRED-MD, the groups would be the predefined variable groups. The random projection forecast we consider, takes a randomly weighted average across the entire predictor set.

Initial convergence rates for ridge regression in factor models as in De Mol et al. (2008) and Carrasco and Rossi (2016) suggested that ridge forecasts may converge at a slower rate compared to PCA-based forecasts. De Mol et al. (2024) show the asymptotic equivalence of PCA-based forecast and ridge regression in settings with possible weak factors. However, this requires additional assumptions that do not fit the factor model of Bai and Ng (2023) that we consider in the present paper. We find that the equivalence between ridge forecasts and PCA-based forecasts holds under serially uncorrelated errors and a crosssection dimension not much larger than the time dimension. However, in other cases the regularization-based methods attain a slower convergence rate.

In a variant of the diffusion index forecast model where the idiosyncratic noise in the factor equation enters the outcome equation through a random effects specification, He (2023) shows the optimality of ridge regression. Importantly, the paper shows the validity of a bias-corrected cross-validation procedure to select the optimal penalty in practice. Our theoretical results determine the optimal penalty parameter up-to-scale, and in practice we set it based on historical forecast performance.

Whereas ridge regression has a long history in forecasting applications, the use of random projections is relatively recent. Random projections are generally motivated from the Johnson-Lindenstrauss lemma (Johnson and Lindenstrauss, 1984), which can be used to show that they maintain a substantial fraction of the information, see for instance Achliop tas (2003). Random projections can be applied to discrete choice models (Chiong and Shum, 2019), to forecast product sales (Schneider and Gupta, 2016), to reduce the dimension of vector autoregressions (Koop et al., 2019), to macroeconomic forecasting (Boot and Nibbering, 2019), and in high-dimensional hypothesis testing (Liu et al., 2023). The equivalence to ridge regression, in terms of the convergence rate, appears new.

Notation Denote ^δNT = min(^√ N, ^√ T) and the jth eigenvalue of a matrix A′A as µ^j (A′A). O(T) denotes the space of orthogonal T × T matrices. Let, ||A||² = tr(A′A) denote the Frobenius norm. The vector b^t is the standard basis vector with a 1 in position t. Throughout, M denotes a generic positive constant independent of any of the involved dimensions. For random variables X and Y , we write X (d) = Y if X and Y have the same distribution. We write T ≍ N if both T/N = O(1) and N/T = O(1).

2 Model and forecasts

We consider the diffusion index forecast model by Stock and Watson (2002a,b) where a small number of factors f^t ∈ R ^r drives the variable of interest, and the same factors underlie a high-dimensional set of predictors x^t ∈ R N ,

$$\boldsymbol{x}_t = \boldsymbol{\Lambda} \boldsymbol{f}_t + \boldsymbol{e}_t, \tag{1}$$

$$y_{t+h} = \mathbf{f}t' \mathbf{\gamma} + \varepsilon{t+h}. \tag{2}$$

Let t = 0, . . . , T, X = (x0, . . . , x^T ⁻h) ′ , F = (f⁰ , . . . , f ^T ⁻^h ) ′ , Λ = (λ1, . . . ,λ^N ) ′ , and y = (yh, . . . , y^T ) ′ . One can add lagged outcome variables to (2), but we omit those from the analysis to keep the discussion focused on the effects of a weak factor structure. Throughout, we write ¯yt+^h for the conditional mean of (2), i.e. ¯yt+^h = f ′ ^tγ.

We follow Bai and Ng (2023) in defining the rescaled matrix Z = X/ NT that has the singular value decomposition

$$Z = UDV' = U_r D_r V'r + U{-r} D{-r} V'{-r}.$$ (3)

Here U ∈ O(T), D is T × N with the δ 2 NT nonzero singular values of Z on the diagonal of its upper left δ 2 NT × δ 2 NT submatrix, and V ∈ O(N). Moreover, D^r ∈ R r×r is a diagonal matrix with the r largest singular values on its diagonal, the columns of U^r ∈ R ^T ^×^r and V ^r ∈ R N×r contain the corresponding left- and right-singular vectors. The matrices U ⁻^r and V ⁻^r contain the remaining left- and right-singular vectors.

The standard assumption on the loadings accompanying (1) is that Λ ′Λ/N →^p Σ^Λ where Σ^Λ is a positive definite matrix. As in Bai and Ng (2023), this paper considers the case where Λ ′Λ/N^α →^p Σ^Λ for some α ∈ (0, 1]. When α = 1, we say that the loadings are 'strong', while when α < 1, we refer to the loadings as 'weak'.

2.1 PCA-based forecasts

PCA-based forecasts rely on the following estimates of the factors and loadings,

$$(\tilde{\boldsymbol{F}}, \tilde{\boldsymbol{\Lambda}}) = (\sqrt{T}\boldsymbol{U}_r, \sqrt{N}\boldsymbol{V}_r\boldsymbol{D}_r). \tag{4}$$

Denote by γˆ the least squares estimator of γ in (2) with f^t replaced by f˜ ^t = F˜ ′ bt . The resulting PCA-based forecast is given by,

$$\hat{y}{T+h|T}^{pca} = \tilde{\boldsymbol{f}}{T}'\hat{\boldsymbol{\gamma}},\tag{5}$$

where

$$\tilde{\boldsymbol{f}}{T} = \frac{1}{\sqrt{N}} \boldsymbol{D}{NT,r}^{-1} \boldsymbol{V}{NT,r}' \boldsymbol{x}{T} = O{p}(1) \frac{1}{N^{\alpha}} \tilde{\boldsymbol{\Lambda}}' \boldsymbol{\Lambda} \boldsymbol{f}{T} + O{p}(1) \frac{1}{N^{\alpha}} \tilde{\boldsymbol{\Lambda}}' \boldsymbol{e}{T}.$$ (6)

Stock and Watson (2002a) Theorem 2 shows that f˜ ′ ^T γˆ − f ′ ^T γ →^p 0 and the limiting distribution was established in Bai and Ng (2006). Under an approximate factor model with strong loadings, the convergence rate is δNT .

Remark There is a slight subtlety in the construction of the forecasts. Suppose that e^t is independent across t. In that case, in the final term of (6), e^T is independent of the error in estimating the loadings. This independence can be used to derive an improved convergence rate. An alternative way to construct the PCA-based forecast is to perform an eigendecomposition on the matrix formed by (x ′ 0 , . . . , x ′ T ). However, in (6), this would lead to dependence between e^T and Λ˜ . With currently available results, this yields a slower convergence rate.

2.2 Regularization-based forecasts

When the number of predictors is large, we can regularize the (potentially ill-defined) inverse of X′X by adding a ridge penalty (Hoerl and Kennard, 1970). This leads to the following forecast.

$$\hat{y}_{T+h|T}^{ridge} = \boldsymbol{x}_t' \left( \boldsymbol{X}' \boldsymbol{X} + \frac{NT}{k} \boldsymbol{I}_N \right)^{-1} \boldsymbol{X}' \boldsymbol{y}. \tag{7}$$

This forecast is identical to a Bayesian forecast with a N(0, k NT I^N ) prior on the regression coefficients. De Mol et al. (2008) analyze the consistency of this forecast for the conditional mean ¯yt+^h and find that it may converge at a slower rate compared to the PCA-based forecast (5), even when loadings are strong. We will show in Section 3 that the convergence rate of (7) actually coincides with that of (5) under strong loadings.

Boot and Nibbering (2019) consider the following alternative forecasting model that is based on random projections of the available variables,

$$y_{t+h,R} = \mathbf{x}_t' \mathbf{R} \boldsymbol{\theta}R + \xi{t+h}, \tag{8}$$

where R ∈ R ^N×^k and the element of R are independent standard normal random variables. We emphasize that (8) is interpreted as a model that defines a forecast. The actual data generating process is always taken to be (1) and (2).

Denote by θˆ ^R the least squares estimator of θ^R in (8) and denote the associated forecast as ˆy rp ^T ⁺h|T,R = x ′ ^TRθ^ˆ ^R. The random projection introduces additional noise in the forecast, which can be reduced by averaging over multiple draws of the random matrix R. The random projection forecast then approximates,

$$\hat{y}_{T+h|T}^{rp} = \boldsymbol{x}_T' \mathbb{E}_R[\boldsymbol{R}\hat{\boldsymbol{\theta}}_R] = \boldsymbol{x}_T' \mathbb{E}_R[\boldsymbol{R}(\boldsymbol{R}'\boldsymbol{X}'\boldsymbol{X}\boldsymbol{R})^{-1}\boldsymbol{R}']\boldsymbol{X}'\boldsymbol{y}. \tag{9}$$

Boot and Nibbering (2019) show that if we take O(N log N) draws of the projection matrix R, the mean squared forecast error approaches that of ˆy rp T +h|T

It is not a coincidence that the regularization in both (7) and (9) is determined by the parameter k, as it is scaled such that the same k (up to a constant scaling factor) determines the convergence rate to the conditional mean. In the subsequent sections, we use the notation ˆy reg T +h|T to refer to both the ridge regression forecast and the random projection forecast.

2.3 Regularizing the empirical eigenvalues

For all three forecasts, the leading term in the expansion of the forecasts that captures the conditional mean f ′ ^T γ can be written as f ′ T (F ′F) ⁻¹F ′U∆U ′F γ. Suppose that we know that the true number of factors is r, and the regularization parameter k is selected as we derive in Section 3. Then, ∆ is a diagonal matrix that satisfies the following.

$$\frac{\Delta_{ii}}{i = 1, \dots, r} \qquad \frac{\text{PCA}}{1} \qquad \frac{\text{Ridge and RP}}{1 - O_p\left(\frac{N}{N^{\alpha}}k^{-1}\right)}$$

$$i = r + 1, \dots, \delta_{NT}^2 \qquad 0 \qquad Op\left(k\delta{NT}^{-2}\right)$$

This clarifies that PCA is hard-thresholding the eigenvalues, while ridge regression and random projections are soft-thresholding the eigenvalues. However, in the asymptotic limit where (N, T) → ∞, ridge regression and random projection may converge to PCA under a suitably chosen of k. To establish this formally requires a more detailed analysis of other terms appearing in the expansion of the forecasts. We also see that for weaker loadings, the regularization parameter k has to be set larger to capture the signal in the first r factors. However, at the same time this leads to a slower decline to zero for the remaining eigenvalues as this part is independent of the loading strength α. This will eventually lower the convergence rate under weak loadings.

3 Theory

We have the following assumptions.

Assumption 1 (i) The errors in (1) can be written as e = R 1/2 ^T aG¹/² ^N , with R^T ∈ R T ×T and G^N ∈ R ^N×^N deterministic matrices,

(ii) Conditional on Λ and F, ait is i.i.d. over i and t, with E[ait|Λ,F] = 0, E[a 4 it|Λ,F] ≤ M,

(iii) $$\mu_T(\mathbf{R}_T) \geq C$$ , $\mu_N(\mathbf{G}_N) \geq C$ ,

(iv) $$\max{t} \sum{s=1}^{T} |\mathbf{R}{T,ts}| \le C$$ , $\max{i} \sum{j=1}^{N} |\mathbf{G}{N,ij}| \le C$ .

This assumption is closely related to the assumptions used in Onatski (2010) and Ahn and Horenstein (2013). Part (iv) slightly strengthens the usual assumption on the maximum eigenvalue of R^T and G^N , so we can easily establish that Assumption 1 implies Assumption A1 and A3 in Bai and Ng (2023).

Corollary 1 Let e^t be defined as before and denote by e(i) the ith column of e. Under Assumption 1 we have the following.

(i) For all $$t$$ , $\frac{1}{N\sqrt{T}} | \boldsymbol{e}_t' \boldsymbol{e}' | = Op(\delta{NT}^{-1})$ and for all $i$ , $\frac{1}{T\sqrt{N}} | \boldsymbol{e}_{(i)}' \boldsymbol{e} | = Op(\delta{NT}^{-1}) |$ ,

(ii) For each $$t$$ , $\mathbb{E}||N^{-\alpha/2}\sum_{i=1}^{N} \lambdai e{it}||^2 \le C$ and $\frac{1}{NT}e_t'e'F = Op(\delta{NT}^{-2})$ ,

(iii) For each $$i$$ , $\mathbb{E}||T^{-1/2}\sum{t=1}^{T} \boldsymbol{f}{t}e{it}||^{2} \leq M$ and $\frac{1}{N^{\alpha}T}\boldsymbol{e}'{(i)}\boldsymbol{e}\boldsymbol{\Lambda} = O{p}\left(\frac{1}{N^{\alpha}}\right) + O{p}\left(\frac{1}{\sqrt{N^{\alpha}T}}\right)$ ,

(iv) $$\mathbf{\Lambda}' \mathbf{e}' \mathbf{F} = \sum{t=1}^{T} \sum{i=1}^{N} \lambda_i \mathbf{f}'t e{it} = O_p(\sqrt{N^{\alpha}T}).$$

We continue with Assumption A2 from Bai and Ng (2023).

Assumption 2 (i) $$\mathbb{E}[|\boldsymbol{f}t|^4] \leq M$$ , $\operatorname{plim}{T \to \infty} \frac{\boldsymbol{F}'\boldsymbol{F}}{T} = \boldsymbol{\Sigma}_F \succ 0$ ,

(ii) $$|\lambdai| \le M$$ , $\lim{N\to\infty} \frac{\Lambda'\Lambda}{N^{\alpha}} = \Sigma_{\Lambda} \succ 0$ for some $\alpha > 0$ with $\alpha \in (0,1]$ ,

(iii) The eigenvalues of ΣΛΣ^F are distinct.

We also repeat Assumption A4 from Bai and Ng (2023) that limits the expansion rate of the cross-section dimension relative to the time dimension.

Assumption 3 As (N, T) → ∞, N N^α 1 ^T → 0, for α as before.

To derive results for the diffusion index model, we need to make assumptions on the regression errors ε^t , which are provided by the following assumption.

Assumption 4 The regression errors ε satisfies E[ε|F, Λ, e] = 0 and E[εε′ |F, Λ, e] = Σ^ε with ∥Σε∥ ≤ M < ∞ a.s.

Finally, several results that we derive can be improved under the following assumption that strengthens Assumption 1 part (i) to ensure that the noise is independent over time.

Assumption 5 In Assumption 1 part (i), set R^T = I.

3.1 Convergence rates for PCA-based forecasts

Under strong factors, Stock and Watson (2002a) show consistency of the PCA forecast (5) for the conditional mean ¯y^T ⁺h|^T = f ′ ^T γ when (N, T) → ∞. Bai and Ng (2006) further show that ˆy pca ^T ⁺h|^T − y¯^T ⁺h|^T = Op(δ −1 NT ).

Under weaker loadings, we have the following result for the PCA forecast.

Theorem 1 Under Assumptions 1 to 4, the PCA-based forecast (5) satisfies

$$\hat{y}{T+h|T}^{pca} - \bar{y}{T+h|T} = O_p \left(\frac{1}{N^{\alpha/2}}\right) + O_p \left(\frac{1}{T^{1/2}}\right) + O_p \left(\frac{N^{3/2}}{N^{3\alpha/2}T}\right) + O_p \left(\frac{N^{1/2}}{N^{3\alpha/2}}\right) + O_p \left(\frac{N^2}{N^{2\alpha}T^{3/2}}\right) + O_p \left(\frac{N}{N^{2\alpha}T^{1/2}}\right).$$ (10)

The first two terms on the right-hand side reduce to the known results when α = 1. However, a number of additional terms appear. To interpret these, we analyze three cases of interest. In the first, the loadings are strong, so α = 1. In the second, the loadings are weaker α < 1, and N ≍ T. In the third setting, again α < 1, but now the time dimension is of lower order relative to the cross-section dimension in the sense that T = O(N^γ ) with γ < 1. Specialized to these cases, Theorem 1 yields the following.

Corollary 2 Case 1. Strong loadings: α = 1.

$$\hat{y}{T+h|T}^{pca} - \bar{y}{T+h|T} = Op(\delta{NT}^{-1}). \tag{11}$$

Case 2. Weaker loadings: α < 1. N ≍ T.

$$\hat{y}{T+h|T}^{pca} - \bar{y}{T+h|T} = O_p \left( \frac{1}{N^{\alpha/2}} \right) + O_p \left( \frac{1}{N^{(3\alpha-1)/2}} \right). \tag{12}$$

Case 3. Weaker loadings: α < 1. T = O(N^γ ), 0 < γ < 1.

$$\hat{y}{T+h|T}^{pca} - \bar{y}{T+h|T} = O_p \left( \frac{1}{N^{\alpha/2}} \right) + O_p \left( \frac{1}{N^{\gamma/2}} \right) + O_p \left( \frac{1}{N^{(3\alpha - 3 + 2\gamma)/2}} \right). \tag{13}$$

Case 1 shows that under strong loadings, we obtain the known result on the convergence rate of PCA-based forecast. Case 2 shows that under weaker loadings, an additional term enters that requires that α > 1/3 for the forecast to remain consistent for the conditional mean. This result is not surprising, as Bai and Ng (2023) already show that the same condition is required for (a rotation of) the factors F to be consistently estimated. We will later show that this restriction can be lifted under Assumption 5. Under Case 3, we see an interaction between the loading strength parameter α and the parameter γ that governs how much smaller T is relative to N. For consistency of the forecasts for the conditional mean, a sufficient condition is now that α > 1 − 3 γ. This indicates that as the crosssectional dimension increases relative to the number of time periods, we require stronger loadings for the forecast to be consistent.

3.2 Convergence rates under regularization

The forecast accuracy of both the ridge-regularized forecast (7) and the random projection forecast (9), depends crucially on the choice of the regularization parameter k. The following theorem stipulates the conditions imposed on k needed for consistency, as well as the resulting convergence rate.

Theorem 2 Let Assumptions 1 to 4 hold. If k is such that as (N, T) → ∞, δ −2 NT k → 0 and ^q N ^N^α k ⁻¹ → 0, then

$$\hat{y}{T+h|T}^{reg} - \bar{y}{T+h|T} = O_p \left( \frac{1}{N^{\alpha/2}} \right) + O_p \left( \frac{1}{T^{1/2}} \right) + O_p \left( \frac{N^{3/2}}{N^{3\alpha/2}T} \right)

O_p \left( \frac{N^{1/2}}{N^{3\alpha/2}} \right) + O_p \left( \frac{N^2}{N^{2\alpha}T^{3/2}} \right) + O_p \left( \frac{N}{N^{2\alpha}T^{1/2}} \right)
O_p \left( \frac{N}{N^{\alpha}}k^{-1} \right) + Op \left( k\delta{NT}^{-1} \left[ \sqrt{\frac{N}{N^{\alpha}}\frac{1}{T}} + \sqrt{\frac{1}{N^{\alpha}}} + \frac{1}{N^{\alpha/4}T^{1/4}} \right] \right).$$ (14)

The first two lines are identical to those obtained for the PCA-based forecast. The second line shows how the regularization parameter balances capturing the signal from the true factors while suppressing the noise coming from the idiosyncratic errors. Again, we specialize this result to three special cases considered before.

Corollary 3 Case 1. Strong loadings: α = 1. Set the regularization parameter as k = O((Nδ² NT ) 1/4 ).

$$\hat{y}{T+h|T}^{reg} - \bar{y}{T+h|T} = Op(\delta{NT}^{-1}). \tag{15}$$

Case 2. Weaker loadings: α < 1. N ≍ T. Set the regularization parameter as k = O(N(3−α)/⁴ ).

$$\hat{y}{T+h|T}^{reg} - \bar{y}{T+h|T} = O_p \left( \frac{1}{N^{\alpha/2}} \right) + O_p \left( \frac{1}{N^{(3\alpha-1)/4}} \right). \tag{16}$$

Case 3. Weaker loadings: α < 1. T = O(N^γ ), 0 < γ < 1. Set the regularization parameter as k = O(N(1+2γ−α)/⁴ ). Then,

$$\hat{y}{T+h|T}^{reg} - \bar{y}{T+h|T} = O_p \left( \frac{1}{N^{\alpha/2}} \right) + O_p \left( \frac{1}{N^{\gamma/2}} \right) + O_p \left( \frac{1}{N^{(3\alpha - 3 + 2\gamma)/4}} \right). \tag{17}$$

In Case 1, we obtain the convergence rate δ −2 NT . This is identical to PCA and improves the convergence rate presented in De Mol et al. (2008) for ridge regression. In Case 2 with weaker loadings and N ≍ T, Corollary 3 shows that for consistency, it is again sufficient that α > 1/3. However, the second term under the optimal choice of the regularization parameter k decays slower to zero compared to what is the case for PCA-based fore-

Figure 1: Convergence rates for PCA-based forecast and regularization-based forecasts

The figure presents the convergence rate achieved for different values of the parameter α that governs the strength of the loadings. The solid line shows the convergence rate for PCA-based forecasts, the dashed line for ridge regression and random projection forecasts. The left panel shows the case where N ≍ T, the right panel the case where T = O(N^γ ) for γ ∈ {0.1, 0.5, 0.9}.

casts. Figure 1 plots the different convergence rates for the PCA-based forecast and the regularization-based forecasts as a function of the strength of the loadings.

In Case 3, when T = N^γ for γ < 1, we reach a similar conclusion as for Case 2: the conditions under which the regularization-based forecasts are consistent are the same as for PCA, but regularization results in a lower convergence rate. The right panel of Figure 1 graphically shows the difference in the convergence rate for γ = {0.1, 0.5, 0.9}.

3.3 Faster convergence rates under serially uncorrelated idiosyncratic errors

The results of the previous section show that in general, we expect the regularization methods to suffer under weaker loadings. However, we now show that a substantial improvement over the convergence rates established in Theorem 1 and Theorem 2 can be obtained by introducing Assumption 5, under which the rows of the idiosyncratic error term are independent. In this case, we have the following.

Theorem 3 Under Assumptions 1 to 5, the PCA-based forecast (5) satisfies

$$\hat{y}{T+h|T}^{pca} - \bar{y}{T+h|T} = O_p \left( \frac{1}{N^{\alpha/2}} \right) + O_p \left( \frac{1}{T^{1/2}} \right). \tag{18}$$

We see that the condition that α > 1/3 is no longer necessary. What is still required is Assumption 3. Case 3, where T = O(N^γ ), therefore requires α + γ > 1.

The results for ridge regression and random projection forecasts can be similarly improved to the following.

Theorem 4 Under Assumptions 1 to 5, and setting k such that as (N, T) → ∞, δ −2 NT k → 0 and ^q N ^N^α k ⁻¹ → 0, the regularization-based forecast (9) satisfies

$$\hat{y}{T+h|T}^{reg} - \bar{y}{T+h|T} = O_p\left(\frac{1}{N^{\alpha/2}}\right) + O_p\left(\frac{1}{T^{1/2}}\right) + O_p\left(\frac{N}{N^{\alpha}}k^{-1}\right) + Op\left(k \cdot \delta{NT}^{-2}\right). \tag{19}$$

For Case 1 with k = O(δNT ) and Case 2 with k = O(N¹−α/² ), we now obtain the same result as presented for PCA in Theorem 3,

$$\hat{y}{T+h|T} - \bar{y}{T+h|T} = O_p \left(\frac{1}{N^{\alpha/2}}\right) + O_p \left(\frac{1}{T^{1/2}}\right). \tag{20}$$

However, in Case 3, when T = N^γ for 0 < γ < 1, and k = O(N(1−α+γ)/² ), we obtain

$$\hat{y}{T+h|T}^{reg} - \bar{y}{T+h|T} = O_p \left( \frac{1}{N^{\alpha/2}} \right) + O_p \left( \frac{1}{N^{\gamma/2}} \right) + O_p \left( \frac{1}{N^{(\alpha+\gamma-1)/2}} \right). \tag{21}$$

The requirement for consistency is α + γ > 1 as it is for PCA. However, in some parts of the parameter space, especially if both α and γ are substantially below 1, the convergence rate is lower compared to PCA. The convergence rates are shown in Figure 2 for different values of γ and α.

Figure 2: Convergence rates under time independent noise and T = O(N^γ )

The figure presents the convergence rate achieved for different values of the parameter α that governs the strength of the loadings and the parameter γ that determines the expansion rate of T /N^γ = O(1).

4 Simulations

We consider the model (1) and (2) with r = 2 factors. The elements of the factors F are independent standard normal random variables. To tune the strength of the loadings, we generate [Λ]ij = q N^α N Zij with Zij standard normal independent over i = 1, . . . , N and j = 1, . . . , r. The parameter α will be varied over {0.5, 0.75, 1} where α = 1 corresponds to a setting with strong factors. We first consider a setting with the idiosyncratic noise generated as eti ∼ N(0, r), independently across t = 1, . . . , T and i = 1, . . . , N. We subsequently consider a setting with serial correlation in the idiosyncratic noise.

In the forecast equation (2), the parameter γ = (1, 1)′ and ε^t ∼ N(0, r) independently over t. We consider one-step ahead forecasts and set ε^T +1 = 0 as this is a noise term that we cannot forecast. We generate PCA-based forecasts assuming that the number of factors is known. For ridge regression and random projections, we pick k ex post to minimize the mean squared forecast error. We compare the forecasts in terms of their Mean Squared Forecast Error (MSFE) defined as E[(ˆy^T ⁺h|^T − y¯^T ⁺h|^T ) 2 ]. Finally, we vary N and T over the grid {100, 200, 300, 400, 500}. Results are averaged over 5,000 realizations of the data generating process.

= 0.5 100 200 300 400 500 0 0.2 0.4 0.6 0.8 MSFE PCA RP Ridge 100 200 300 400 500 0 20 40 60 80 100 Regularization (k) PCA RP Ridge = 0.7 100 200 300 400 500 0 0.2 0.4 0.6 0.8 MSFE 100 200 300 400 500 0 20 40 60 80 100 Regularization (k) = 1 100 200 300 400 500 0 0.2 0.4 0.6 0.8 MSFE 100 200 300 400 500 0 20 40 60 80 100 Regularization (k)

Figure 3: Mean squared error and regularization parameter (N = T)

The figure presents the Mean Squared Forecast Error (MSFE) Eˆ[(ˆy^T +1|^T −y¯^T +1|^T ) 2 ] in the left panel, with Eˆ indicating the average over 5,000 Monte Carlo simulations. The data generating process accords with (1) and (2) with r = 2 factors, and the elements of the factors, loadings and idiosyncratic noise terms are all independent standard normals. We compare the PCA-based forecast, the ridge regression forecast and the random projection forecast. For the latter two, the choice of k is shown in the right panel. We set T = N and consider α = 0.5 (upper panel), α = 0.75 (middle panel) and α = 1 (lower panel).

N and T

4.1 Independent idiosyncratic noise

N and T

The left panel of Figure 3 shows the one-step ahead MSFE for PCA-based forecasts as well as ridge and random projection forecasts when we set T = N. The right panel of the figure displays the optimal choice of the regularization parameter for the latter two methods. We observe the following: regardless of the value of α and the values of N and T, PCAbased forecasts offer the lowest MSFE. There is no observable difference between ridge and random projection forecasts in terms of the MSFE. We also see that as the loadings become

Figure 4: MSFE and regularization parameter (varying T and N)

The figure presents the Mean Squared Forecast Error (MSFE) Eˆ[(ˆy^T +1|^T − y¯^T +1|^T ) 2 ] for ridge forecasts relative to the MSFE of the PCA-based forecasts. Top panel has weak loadings with α = 0.75, bottom panel has strong loadings with α = 1. In the left panel, we fix T at {200, 300, 400, 500} and vary N on the x-axis. In the right panel, we fix N at {200, 300, 400, 500} and vary T on the x-axis. For further details, see the note following Figure 3.

weaker, the difference between PCA and the regularization methods appears to increase. With regard to the regularization parameter, we see that it increases with increasing (N, T) and with weaker loadings.

Figure 4 shows the MSFE of ridge regression relative to PCA-based forecasts. We omit the random projection forecasts, because the accuracy is identical to the ridge regression forecasts. In the left panel of Figure 4, we fix T and vary N on the x-axis. In the right panel, we fix N and vary T on the x-axis. The upper panel has weak loadings with α = 0.75, while the lower panel has strong loadings with α = 1. The most important finding is that for strong loadings, the relative performance between regularization-based methods and PCA stabilizes as either N or T increases. This confirms that the convergence rates for these methods are identical under strong loadings. Under weaker loadings (upper panel), we see that the relative MSFE stabilizes with increasing T (right panel), yet it

= 0.5 100 200 300 400 500 0 0.2 0.4 0.6 0.8 MSFE PCA RP Ridge PCA(k) 100 200 300 400 500 0 20 40 60 80 100 Regularization (k) PCA RP Ridge PCA(k) = 0.7 100 200 300 400 500 0 0.2 0.4 0.6 0.8 MSFE 100 200 300 400 500 0 20 40 60 80 100 Regularization (k) = 1 100 200 300 400 500 N and T 0 0.2 0.4 0.6 0.8 MSFE 100 200 300 400 500 N and T 0 20 40 60 80 100 Regularization (k)

Figure 5: MSFE and regularization parameter with serially correlated errors

Details are in the note following Figure 3, with the addition that the idiosyncratic noise follows an AR(1) process with autocorrelation parameter ρ = 0.7. PCA(k) denotes a PCA-based forecast with ex post optimal selection of the number of factors.

continues to increase with larger N (left panel). This confirms the theoretical results from Section 3.3 that state that under weaker loadings, the regularization methods may attain a lower convergence rate when N is increasing at a faster rate compared to T.

4.2 Serially correlated idiosyncratic errors

We change the setup by introducing serial correlation in the idiosyncratic component of the predictors. To be precise, we consider Eti = ρEt−1,i + vt−1,i with vti ∼ N(0, 1 − ρ 2 ) independent over t and i, E1ⁱ = v1ⁱ ∼ N(0, 1) and ρ = 0.7.

Figure 5 shows again the MSFE and the optimal choice for the regularization parameter, akin to Figure 3. There are two important observations. First, we see a noticeable deterioration in the performance of PCA-based forecasts under weak factors and small sample sizes. There is hardly any change in the MSFE for the ridge or random projection forecasts, and we only find a slightly smaller optimal value of k. The deterioration of PCA can be explained intuitively by the fact that the eigenvalues are no longer clearly separated from the largest eigenvalue of the idiosyncratic errors. Suppose that the three largest eigenvalues are of the same order of magnitude. Because the autocorrelation increases the maximum eigenvalue of the covariance matrix of the idiosyncratic noise, it may be the case that the second largest eigenvalue is associated with a singular vector that is not related to the unobserved factors, while the third largest eigenvalue is associated with the unobserved factors. In this case, PCA would miss the relevant signal as it hard thresholds the eigenvalues at the supposed true number of factors r. The regularization-based methods, on the other hand, will apply comparable soft-thresholding to the largest three eigenvalues, maintaining the majority of the signal and thereby improving over PCA.

Figure 5 also adds PCA(k), which selects the optimal number of factors ex post. We see that indeed for weak loadings, the optimal number of factors is much larger than the nominal number of factors. Moreover, choosing the optimal number of factors largely mitigates the deterioration of the PCA-based forecasts. For small sample sizes, and weak factors, the regularization-based methods continue to offer more accurate forecasts. The second important observation is however that when α = 0.5 and we use PCA(k), we see that the MSFE for PCA converges faster to zero relative to random projections and ridge. This confirms the theoretical results in Corollary 3 Case 2 versus Corollary 2 Case 2.

5 Application: forecast accuracy on monthly and quarterly macroeconomic data sets

To confirm the theoretical results in practice, we setup a forecasting exercise to assess the impact of the ratio T/N on forecasts from PCA, ridge and random projections in an empirical setting. The theory suggests that as T/N decreases, PCA becomes relatively more accurate. However, the simulations also show that in small samples, the regularization methods may yield more accurate forecasts. Our goal is therefore analyse prediction accuracy on variables from the FRED-MD (McCracken and Ng, 2016) and FRED-QD (McCracken and Ng, 2020). These data sets are large cross-sections of macroeconomic and financial variables and have been studied using PCA and random projections (Boot and Nibbering, 2019). We select vintage 2024-01 for both data sets.

For each variable, forecasts are generated from the model

$$y_{t+h} = \boldsymbol{w}_t' \boldsymbol{\beta}_w + \boldsymbol{x}_t' \boldsymbol{R} \boldsymbol{\beta}x + \varepsilon{t+h}, \tag{22}$$

where the N^w × 1 vector w^t are the first four lags of y^t and an intercept, which are always included. The Nx×1 vector x^t are the fifth and sixth lag of the target variables, and all the other variables. The N^x ×k matrix R with k < N^x reduces the dimension of the regressors.

The sample runs from 1959M1 (or 1959Q1) up to the end of 2023. Forecasting starts in the first period of 1970, i.e. 1970M1 for the monthly set and 1970Q1 for quarterly data. Out-of-sample evaluation begins in 1980M1 or 1980Q1, after a 10-year burn-in to select the hyperparameters that minimize the squared forecast error. So, the number of factors, the subspace dimension, and the ridge penalty are optimized over an expanding window that starts in 1970. The maximum number of factors in PCA and the subspace dimension in RP are set to 50. Random projection forecasts are averaged over 1,000 draws. For ridge, we set up a grid for the log of the penalty parameter that runs from -14.7 to 15, with increments of 0.3, following Boot and Nibbering (2019).

The figure presents the percentage of times that forecasts from PCA has a smaller mean squared forecast error (MSFE) loss compared to forecasts using ridge (blue) or random projections (RP, orange) out of 127 variables from the FRED-MD. The fraction T /N^x changes from varying moving window lengths while N^x is fixed. The evaluation sample is 1980M1–2023M12.

We conduct two experiments to assess the impact of the ratio T/Nx, changing the length or frequency of the estimation sample while keeping the number of regressors, Nx, fixed.

5.1 Moving window size

First, we vary the size of the moving window, T, from ¹ ³N^x to ⁴ ³Nx. The advantage of comparing across moving windows is that it controls the ratio T/N^x as it is fixed over the evaluation sample. Monthly data allows for a wider range of moving windows and a sufficient number of forecasts for evaluation.

Figure 6 presents the percentage of times that PCA-based forecasts achieve a smaller MSFE compared to ridge or random projection forecasts. The percentages are below 50 percent for all window sizes, indicating that ridge and random projections yield on average more accurate forecasts than PCA in this dataset. For shorter moving window lengths, the forecasts obtained by PCA improve in terms of MSFE relative to ridge and random projections. With a window length of T = ⁶Nx, PCA yields better forecasts in about 10

Figure 7: DM test statistics

The figure presents box plots of DM test statistics for the difference in squared error loss between ridge (left panel) or random projections (right panel) forecasts compared to PCA-based forecasts. The fraction T /N^x changes from varying moving window lengths while N^x is fixed. Black dots represent the mean DM statistic. Light gray dashed lines correspond to the two-sided 5% critical value, assuming a standard normal distribution. The evaluation sample is 1980M1–2023M12.

percent of the cases, while PCA-based forecasts are more accurate than ridge and random projections for close to 40 percent of the variables when T = 1 ³Nx. The increase in relative performance between longer and shorter windows is not monotonic, but there is a clear pattern indicating that ridge and random projections yield relatively less accurate forecasts than PCA at shorter windows.

Figure 7 presents the distribution of Diebold-Mariano (DM) test statistics (Diebold and Mariano, 1995) computed on the difference between the squared error loss from regularizationand PCA-based forecasts. Positive values correspond to more accurate forecasts by ridge or random projections. The distribution of DM statistics is shifted downwards for shorter windows relative to the distribution for longer windows. DM statistics are more often negative with shorter windows, see also Figure 6. When forecasting CPI: Durables, PCA even produces significantly more accurate forecasts than ridge or random projections with windows shorter than T = ⁶Nx, while the DM statistics are positive with windows larger than T = Nx.

Figure 8 presents the difference in DM test statistics between the largest window (T = 4 ³Nx) and the shortest window (T = 1 ³Nx), for each of the variables in the FRED-MD.

Figure 8: Change in DM test statistic (a) Ridge

The figure presents change in DM test statistic across estimation windows for each variable in the FRED-MD. The y-axis shows the DM test statistics from the longest estimation window (T = 4/3Nx) minus the DM test statistic from the shortest window (T = 1/3Nx), where the DM test statistic is computed on the squared forecast loss differential between ridge (top panel) or random projections (bottom panel) versus PCA. The evaluation sample is 1980M1–2023M12. Numbers on the x-axis denote the FRED-MD groups: (1) Output and income; (2) Labor market; (3) Housing; (4) Consumption, orders, and inventories; (5) Money and credit; (6) Interest and exchange rates; (7) Prices; (8) Stock market.

Positive values indicate a shift of relative improvement for PCA compared to ridge or random projections with the shortest window compared to the larger window. The shift in the distribution of DM test statistics is not driven by a particular set of variables because the change is positive for most variables. The shift is particularly large for variables in categories 1 (Output and income) and 7 (Prices) and results in a change in the preferred model. For example, the DM statistic from comparing ridge (random projections) with PCA in the case of IP: Final Products and Nonindustrial Supplies is 1.30 (1.31) with T = 4 ³Nx, while it is -1.40 (-1.63) with T = 4 ³Nx. As an exception, PCA perform relatively worse for variables in category 3 (Housing).

Figure 9: DM test statistics across data frequency

The figure presents box plots of DM test statistics for the difference in squared error loss between ridge (left panel) or random projections (right panel) forecasts compared to PCA-based forecasts. The fraction T /N^x changes from varying the frequency while N^x is fixed. Black dots represent the mean DM statistic. Light gray dashed lines correspond to the two-sided 5% critical value, assuming a standard normal distribution. The evaluation sample is 1980M1–2023M12.

5.2 Frequency

As a second experiment, we compare the FRED-MD and FRED-QD data sets using an estimation window covering the same calendar period but at different frequencies, thus varying T/Nx. We employ a moving window of 10 years, such that the number of observations is T = 120 for monthly and T = 40 for quarterly data. We consider the subset of 102 variables that are in both the FRED-MD and the FRED-QD, see Appendix C.

Monthly observations are aggregated to the quarterly level and monthly forecasts are at a h = 3 month horizon. The quarterly forecasts are at a h = 1 quarter horizon. Hence, the observations and forecasts measure the same entity across the data sets, only at different frequencies.

In 19.6 percent of the cases, PCA has a smaller MSFE than ridge in the quarterly data set and only in 12.7 percent of the cases when using monthly data. Similarly, PCA leads to more accurate forecasts than random projections with 18.6 percent of the variables for quarterly and 5.9 percent for monthly data.

This corresponds to a shift in DM statistics between using data at the quarterly or monthly frequency, for both ridge and random projections, see Figure 9. Using quarterly data results in DM statistics that are smaller than when using monthly data, implying that forecasts are preferred more often when using quarterly data. The change of the DM statistic can be substantial and lead to a different conclusion on which model yields more accurate forecasts. For example, the DM statistic for ridge (random projections) versus PCA is 3.32 (3.24) based on monthly data and -2.23 (-2.10) on quarterly data when forecasting the Switzerland / U.S. Foreign Exchange Rate.

Hence, the empirical findings are consistent between both experiments. Ridge and random projections generally yield better forecasts in the FRED-MD and FRED-QD than PCA. However, when the estimation sample becomes smaller compared to the crosssectional size, i.e. the fraction T/N^x decreases, the accuracy of PCA-based forecasts improves relative to the accuracy of regularization-based forecasts.

6 Conclusion

We analyze the convergence rates of PCA-based forecasts in the diffusion index forecast model with weaker loadings. We compare these rates with forecasts based on ridge regression and random projections. We find that under strong loadings, all forecasts attain the same convergence rate. Under weaker loadings all forecasts are consistent under the same assumptions on the loading strength, but ridge regression and random projection forecasts converge more slowly relative to PCA when the cross-section dimension is large relative to the number of time periods. We confirm these results in simulations and an empirical study based on the FRED-MD and FRED-QD data sets. These numerical results also show that in small samples, regularization-based methods may be more robust to serial correlation in the idiosyncratic errors.

Funding details Tom Boot acknowledges financial support by the Dutch Research Council (NWO) as part of grant VI.Veni.201E.11.

Disclosure statement There are no competing interests to declare.

References

Achlioptas, D. (2003). Database-friendly random projections: Johnson-Lindenstrauss with binary coins. Journal of Computer and System Sciences, 66(4):671–687.
Ahn, S. C. and Horenstein, A. R. (2013). Eigenvalue ratio test for the number of factors. Econometrica, 81(3):1203–1227.
Bai, J. and Ng, S. (2006). Confidence intervals for diffusion index forecasts and inference for factor-augmented regressions. Econometrica, 74(4):1133–1150.
Bai, J. and Ng, S. (2023). Approximate factor models with weaker loadings. Journal of Econometrics, 235(2):1893–1916.
Boot, T. and Nibbering, D. (2019). Forecasting using random subspace methods. Journal of Econometrics, 209(2):391–406.
Brinkhuis, J., Luo, Z.-Q., and Zhang, S. (2005). Matrix convex functions with applications to weighted centers for semidefinite programming. Technical Report No. EI 2005-38, Econometric Institute, Erasmus University Rotterdam.
Carrasco, M. and Rossi, B. (2016). In-sample inference and forecasting in misspecified factor models. Journal of Business & Economic Statistics, 34(3):313–338.
Chiong, K. X. and Shum, M. (2019). Random projection estimation of discrete-choice models with large choice sets. Management Science, 65(1):256–271.
De Mol, C., Giannone, D., and Reichlin, L. (2008). Forecasting using a large number of predictors: Is Bayesian shrinkage a valid alternative to principal components? Journal of Econometrics, 146(2):318–328.
De Mol, C., Giannone, D., and Reichlin, L. (2024). The asymptotic equivalence of ridge and principal component regression with many predictors. Econometrics and Statistics.
Diebold, F. X. and Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13(3):253–263.
Fan, J. and Liao, Y. (2022). Learning latent factors from diversified projections and its applications to over-estimated and weak factors. Journal of the American Statistical Association, 117(538):909–924.
He, Y. (2023). Ridge regression under dense factor augmented models. Journal of the American Statistical Association, 119(546):1566–1578.
Hoerl, A. E. and Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1):55–67.
Johnson, W. B. and Lindenstrauss, J. (1984). Extensions of Lipschitz mappings into a Hilbert space. Contemporary Mathematics, 26(189-206):1.
Johnstone, I. M. and Lu, A. Y. (2009). On consistency and sparsity for principal components analysis in high dimensions. Journal of the American Statistical Association, 104(486):682–693.
Karabiyik, H. and Westerlund, J. (2021). Forecasting using cross-section average– augmented time series regressions. Econometrics Journal, 24(2):313–333.
Koop, G., Korobilis, D., and Pettenuzzo, D. (2019). Bayesian compressed vector autoregressions. Journal of Econometrics, 210(1):135–154.
Liu, C., Zhao, X., and Huang, J. (2023). A random projection approach to hypothesis tests in high-dimensional single-index models. Journal of the American Statistical Association, 119(546):1008–1018.
McCracken, M. and Ng, S. (2020). FRED-QD: A quarterly database for macroeconomic research. Working Paper 26872, National Bureau of Economic Research.
McCracken, M. W. and Ng, S. (2016). FRED-MD: A monthly database for macroeconomic research. Journal of Business & Economic Statistics, 34(4):574–589.
Onatski, A. (2010). Determining the number of factors from empirical distribution of eigenvalues. Review of Economics and Statistics, 92(4):1004–1016.
Onatski, A. (2012). Asymptotics of the principal components estimator of large factor models with weakly influential factors. Journal of Econometrics, 168(2):244–258.
Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica, 17(4):1617–1642.
Schneider, M. J. and Gupta, S. (2016). Forecasting sales of new and existing products using consumer reviews: A random projections approach. International Journal of Forecasting, 32(2):243–256.
Stock, J. H. and Watson, M. W. (1998). Diffusion indexes. Working Paper 6702, National Bureau of Economic Research.
Stock, J. H. and Watson, M. W. (2002a). Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association, 97(460):1167–1179.
Stock, J. H. and Watson, M. W. (2002b). Macroeconomic forecasting using diffusion indexes. Journal of Business & Economic Statistics, 20(2):147–162.
Uematsu, Y. and Yamagata, T. (2022a). Estimation of sparsity-induced weak factor models. Journal of Business & Economic Statistics, 41(1):213–227.
Uematsu, Y. and Yamagata, T. (2022b). Inference in sparsity-induced weak factor models. Journal of Business & Economic Statistics, 41(1):126–139.

Appendix A Proofs

A.1 Preliminary results: some matrix algebra

The following identity appears for example in Brinkhuis et al. (2005). For completeness, we provide a proof below.

Lemma 1

$$(t\boldsymbol{Y}{1} + (1-t)\boldsymbol{Y}{2})^{-1} = t\boldsymbol{Y}{1}^{-1} + (1-t)\boldsymbol{Y}{2}^{-1} - t(1-t)(\boldsymbol{Y}{1}^{-1} - \boldsymbol{Y}{2}^{-1})(t\boldsymbol{Y}{2}^{-1} + (1-t)\boldsymbol{Y}{1}^{-1})^{-1}(\boldsymbol{Y}{1}^{-1} - \boldsymbol{Y}{2}^{-1}).$$ (A.1)

Proof: First rewrite

$$tY_1 + (1-t)Y_2 = Y_1(Y_1^{-1} + t(Y_2^{-1} - Y_1^{-1})Y_2)$$

$$= Y_1(tY_2^{-1} + (1-t)Y_1^{-1})Y_2.$$ (A.2)

Then,

$$(tY{1} + (1-t)Y{2})^{-1}$$

$$= Y{2}^{-1}(tY{2}^{-1} + (1-t)Y{1}^{-1})^{-1}Y{1}^{-1}$$

$$= Y{2}^{-1} - Y{2}^{-1} + Y{2}^{-1}(tY{2}^{-1} + (1-t)Y{1}^{-1})^{-1}Y{1}^{-1}$$

$$= Y{2}^{-1} - Y{2}^{-1}(tY{2}^{-1} + (1-t)Y{1}^{-1})^{-1}(tY{2}^{-1} + (1-t)Y{1}^{-1}) + Y{2}^{-1}(tY{2}^{-1} + (1-t)Y{1}^{-1})^{-1}Y{1}^{-1}$$

$$= Y{2}^{-1} - tY{2}^{-1}(tY{2}^{-1} + (1-t)Y{1}^{-1})^{-1}(Y{2}^{-1} - Y{1}^{-1})$$

$$= Y{2}^{-1} + t(Y{1}^{-1} - Y{2}^{-1}) - t(Y{1}^{-1} - Y{2}^{-1}) - tY{2}^{-1}(tY{2}^{-1} + (1-t)Y{1}^{-1})^{-1}(Y{2}^{-1} - Y{1}^{-1})$$

$$= Y{2}^{-1} + t(Y{1}^{-1} - Y{2}^{-1}) - t(tY{2}^{-1} + (1-t)Y{1}^{-1} - Y{2}^{-1})(tY{2}^{-1} + (1-t)Y{1}^{-1})^{-1}(Y{1}^{-1} - Y{2}^{-1})$$

$$= tY{1}^{-1} + (1-t)Y{2}^{-1} - t(1-t)(Y{1}^{-1} - Y{2}^{-1})(tY{2}^{-1} + (1-t)Y{1}^{-1})^{-1}(Y{1}^{-1} - Y{2}^{-1}).$$ (A.3)

This completes the proof. ■

Lemma 2 Consider a symmetric positive definite matrix Y and a fixed vector x. The function

$$f(\mathbf{Y}) = \frac{\mathbf{x}' \mathbf{Y}^{-1} \mathbf{x}}{1 + \mathbf{x}' \mathbf{Y}^{-1} \mathbf{x}}$$ (A.4)

is matrix convex in Y .

Proof. For matrix convexity, we need to show that for all 0 ≤ t ≤ 1,

$$\frac{x'(tY_1 + (1-t)Y_2)^{-1}x}{1 + x'(tY_1 + (1-t)Y_2)^{-1}x} \le t \frac{x'Y_1^{-1}x}{1 + x'Y_1^{-1}x} + (1-t)\frac{x'Y_2^{-1}x}{1 + x'Y_2^{-1}x}.$$ (A.5)

For notational convenience, define A = (tY ¹ + (1 − t)Y ²) −1 for the moment. Then, (A.5) is equivalent to

$$x'Ax(1+x'Y{1}^{-1}x)(1+x'Y{2}^{-1}x)$$

$$\leq tx'Y{1}^{-1}x(1+x'Y{2}^{-1}x) + (1-t)x'Y{2}^{-1}x(1+x'Y{1}^{-1}x)$$

$$+ x'Ax\left[tx'Y{1}^{-1}x(1+x'Y{2}^{-1}x) + (1-t)x'Y{2}^{-1}x(1+x'Y{1}^{-1}x)\right].$$ (A.6)

Gathering the terms with x ′Ax and simplifying, we get

$$x'Ax(1+(1-t)x'Y_1^{-1}x+tx'Y_2^{-1}x) \le tx'Y_1^{-1}x+(1-t)x'Y_2^{-1}x+x'Y_1^{-1}xx'Y_2^{-1}x.$$ (A.7)

Define B = (Y −1 ¹ −Y −1 2 )(tY −1 ² + (1−t)Y −1 1 ) −1 (Y −1 ¹ −Y −1 2 ). Now using Lemma 1, we get

$$-t(1-t)x'Bx + (1-t)t(x'Y_1^{-1}x)^2 + (1-t)^2(x'Y_1^{-1}x)(x'Y_2^{-1}x)$$

$$-t(1-t)^2x'Bxx'Y_1^{-1}x$$

$$+t^2x'Y_1^{-1}xx'Y_2^{-1}x + t(1-t)(x'Y_2^{-1}x)^2 - t^2(1-t)x'Bxx'Y_2^{-1}x$$

$$\leq x'Y_1^{-1}xx'Y_2^{-1}x.$$ (A.8)

Rewriting some more, we see that this is equivalent to

$$(x'(Y_1^{-1} - Y_2^{-1})x)^2 \le x'Bx(1 + (1-t)x'Y_1^{-1}x + tx'Y_2^{-1}x).$$ (A.9)

Since x ′Bx ≥ 0, it is sufficient to show that

$$(\boldsymbol{x}'(\boldsymbol{Y}{1}^{-1} - \boldsymbol{Y}{2}^{-1})\boldsymbol{x})^{2} \leq \boldsymbol{x}'(\boldsymbol{Y}{1}^{-1} - \boldsymbol{Y}{2}^{-1})[(1-t)\boldsymbol{Y}{1}^{-1} + t\boldsymbol{Y}{2}^{-1})]^{-1}(\boldsymbol{Y}{1}^{-1} - \boldsymbol{Y}{2}^{-1})\boldsymbol{x}$$

$$\cdot \boldsymbol{x}'[(1-t)\boldsymbol{Y}{1}^{-1} + t\boldsymbol{Y}{2}^{-1})]\boldsymbol{x}.$$ (A.10)

However, this inequality follows immediately from the Cauchy-Schwarz inequality, as we can write the left-hand side as

$$(\mathbf{x}'(\mathbf{Y}{1}^{-1} - \mathbf{Y}{2}^{-1})\mathbf{x})^{2} = (\mathbf{x}'(\mathbf{Y}{1}^{-1} - \mathbf{Y}{2}^{-1})[(1-t)\mathbf{Y}{1}^{-1} + t\mathbf{Y}{2}^{-1})]^{-1/2}[(1-t)\mathbf{Y}{1}^{-1} + t\mathbf{Y}{2}^{-1})]^{1/2}\mathbf{x})^{2}.$$ (A.11)

This completes the proof. ■

A.2 Preliminary results: expectations over random projections

Lemma 3 Let R ∈ R N×k be a matrix with independent entries Rij ∼ N(0, 1). Consider a matrix Z ∈ R ^T ^×^N with singular value decomposition Z = UDV ′ . Then,

$$\mathbb{E}{R}[\boldsymbol{R}(\boldsymbol{R}'\boldsymbol{Z}'\boldsymbol{Z}\boldsymbol{R})^{-1}\boldsymbol{R}'] = \boldsymbol{V}\mathbb{E}{R}[\boldsymbol{R}(\boldsymbol{R}'\boldsymbol{D}'\boldsymbol{D}\boldsymbol{R})^{-1}\boldsymbol{R}']\boldsymbol{V}'. \tag{A.12}$$

Proof. Define R˜ = V ′R. Using that V V ′ = I, we have that,

$$R(R'VD'DV'R)^{-1}R' = V\tilde{R}(\tilde{R}'D'D\tilde{R})^{-1}\tilde{R}'V'. \tag{A.13}$$

Since the elements of R are i.i.d. standard normal random variables, so are the elements of R˜ . Redefining R˜ as R gives the result. ■ Lemma 4 Consider the notation as in Lemma 3 and let bⁱ ∈ R ^N be a vector with value 1 at the ith position and zeros elsewhere. Then, b ′ ⁱER[R(R′D′DR) ⁻¹R′ ]b^j = 0 for i ̸= j.

Proof. Since the elements of R are i.i.d. standard normal, R (d) = ZR, where Z is a diagonal matrix that is independent of R and with independent Rademacher random variables on the diagonal. Since D′D is diagonal,

$$b_i' \mathbb{E}_R[\mathbf{R}(\mathbf{R}'\mathbf{D}'\mathbf{D}\mathbf{R})^{-1}\mathbf{R}'] \mathbf{b}_j = bi' \mathbb{E}{Z,R}[\mathbf{Z}\mathbf{R}(\mathbf{R}'\mathbf{D}'\mathbf{D}\mathbf{R})^{-1}\mathbf{R}'\mathbf{Z}'] \mathbf{b}_j$$

$$= b_i' \mathbb{E}_Z[z_i z_j] \mathbb{E}_R[\mathbf{R}(\mathbf{R}'\mathbf{D}'\mathbf{D}\mathbf{R})^{-1}\mathbf{R}'] \mathbf{b}_j = 0.$$ (A.14)

Lemma 5 Let D be an l × l diagonal matrix with elements [D² ]ii = dⁱ such that dⁱ ≥ di+1 and dⁱ ≥ C > 0 for i = 1, . . . , m. Let R a l × k matrix with i.i.d. standard normal entries. Assume that k < m − 2. Consider

$$w_i = e_i' \mathbf{D} \mathbb{E}_R [\mathbf{R} (\mathbf{R}' \mathbf{D}^2 \mathbf{R})^{-1} \mathbf{R}'] \mathbf{D} e_i, \tag{A.15}$$

■

We can bound wⁱ for i = 1, . . . , m as

$$1 - \frac{1}{1 + (k-2)\frac{di}{\sum{j \neq i} d_j}} \le w_i \le 1 - \frac{1}{1 + (d_i/d_m)k/(m-k-2)}$$ (A.16)

Proof. We first show the first inequality. Denote R = [r1, r2, ..., r^l ′ where rⁱ are k × 1 vectors. Define M = R′D²R. Then,

$$\boldsymbol{M} = d_i \boldsymbol{r}_i \boldsymbol{r}i' + \sum{j \neq i} d_j \boldsymbol{r}_j \boldsymbol{r}_j' = d_i \boldsymbol{r}_i \boldsymbol{r}i' + \boldsymbol{M}{-i}. \tag{A.17}$$

By applying the Sherman-Morrison formula and using Lemma 2 and the fact that for any

matrix convex function we have E[h(Z)] ≥ h(E[Z]), we see that

$$\mathbb{E}{R}[d{i}\boldsymbol{r}{i}'\boldsymbol{M}^{-1}\boldsymbol{r}{i}] = \mathbb{E}{R}\left[\frac{d{i}\boldsymbol{r}{i}'\boldsymbol{M}{-i}^{-1}\boldsymbol{r}{i}}{1 + d{i}\boldsymbol{r}{i}'\boldsymbol{M}{-i}^{-1}\boldsymbol{r}{i}}\right] \ \geq \mathbb{E}{r{i}}\left[\frac{d{i}\boldsymbol{r}{i}'\mathbb{E}{R{-i}}[\boldsymbol{M}{-i}]^{-1}\boldsymbol{r}{i}}{1 + d{i}\boldsymbol{r}{i}'\mathbb{E}{R{-i}}[\boldsymbol{M}{-i}]^{-1}\boldsymbol{r}{i}}\right]. \ = \mathbb{E}{r{i}}\left[\frac{\frac{d{i}}{\sum{j\neq i}d{j}}\boldsymbol{r}{i}'\boldsymbol{r}{i}}{1 + \frac{d{i}}{\sum{j\neq i}d{j}}\boldsymbol{r}{i}'\boldsymbol{r}{i}}\right] \ = 1 - \mathbb{E}{r{i}}\left[\frac{1}{1 + \frac{d{i}}{\sum{j\neq i}d{j}}\boldsymbol{r}{i}'\boldsymbol{r}{i}}\right] \ \geq 1 - \frac{\mathbb{E}{r{i}}[(\boldsymbol{r}{i}'\boldsymbol{r}{i})^{-1}]}{\mathbb{E}{r{i}}[(\boldsymbol{r}{i}'\boldsymbol{r}{i})^{-1}] + \frac{d{i}}{\sum{j\neq i}d{j}}} \ = 1 - \frac{1}{1 + (k - 2)\frac{d{i}}{\sum{j\neq i}d{j}}}.$$ (A.18)

For the second inequality in Lemma 5, first notice that M ⪰ d^m P^m ^j=1 rjr ′ ^j ^≡ ^M˜ . This implies that M⁻¹ ⪯ M˜ −1 . Then,

$$\mathbb{E}{R}[d{i}\boldsymbol{r}{i}'\boldsymbol{M}^{-1}\boldsymbol{r}{i}] \leq \mathbb{E}{R}[d{i}\boldsymbol{r}{i}'\tilde{\boldsymbol{M}}^{-1}\boldsymbol{r}{i}] \ = \mathbb{E}{R}\left[\frac{d{i}\boldsymbol{r}{i}\tilde{\boldsymbol{M}}{-i}^{-1}\boldsymbol{r}{i}}{1 + d{i}\boldsymbol{r}{i}\tilde{\boldsymbol{M}}{-i}^{-1}\boldsymbol{r}{i}}\right] \ \leq \frac{d{i}\mathbb{E}{R}\left[\operatorname{tr}\left(\tilde{\boldsymbol{M}}{-i}^{-1}\right)\right]}{1 + d{i}\mathbb{E}{R}\left[\operatorname{tr}\left(\tilde{\boldsymbol{M}}{-i}^{-1}\right)\right]} \ \leq \frac{(d{i}/d{m})k/(m - k - 2)}{1 + (d{i}/d_{m})k/(m - k - 2)}, \tag{A.19}$$

where the last inequality uses the trace of the expectation of a k-dimensional inverse Wishart matrix with m − 1 degrees of freedom. ■

A.3 Preliminary results: factors and loadings

We have the following result quantifying the relation between the true factors and the left singular vectors of Z associated with eigenvalues λr+1, λr+2, . . . , as well as the true loadings and the right singular vectors of Z associated with eigenvalues λr+1, λr+2, . . .

Lemma 6 Denote F, F˜ , U ⁻^r as in the main text. Let H² = (F ′F) ⁻¹F ′F˜ and H³ = (F˜ ′ F/T) −1 . Then,

$$(\mathbf{F}'\mathbf{F})^{-1}\mathbf{F}'\tilde{\mathbf{F}}\tilde{\mathbf{F}}'\mathbf{F}/T = \mathbf{I} + O_p\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) + O_p\left(\frac{1}{N^{\alpha}}\right) + O_p\left(\frac{N}{N^{\alpha}T}\right), \quad (A.20)$$

$$|(\mathbf{F}'\mathbf{F})^{-1/2}\mathbf{F}'\mathbf{U}_{-r}|^2 = O_p\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) + O_p\left(\frac{1}{N^{\alpha}}\right) + O_p\left(\frac{N}{N^{\alpha}T}\right), \tag{A.21}$$

$$|(\mathbf{\Lambda}'\mathbf{\Lambda})^{-1}\mathbf{\Lambda}'\mathbf{V}_{-r}\mathbf{\Lambda}'| = O_p\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) + O_p\left(\frac{1}{N^{\alpha}}\right) + O_p\left(\frac{N}{N^{\alpha}T}\right). \tag{A.22}$$

Proof. The first part is Lemma 3 of Bai and Ng (2023). For (A.21), using the fact that UU′ = I^T , we have

$$F'F = F'UU'F$$

$$= F'U_rU'rF + F'U{-r}U'_{-r}F$$

$$= F'\tilde{F}\tilde{F}'F/T + F'U{-r}U'{-r}F$$

$$= (F'F)H_2H3^{-1} + F'U{-r}U'_{-r}F.$$

Multiplying from the left by (F ′F) ⁻¹ and using Lemma 3 of Bai and Ng (2023), we obtain

$$\begin{split} |(\mathbf{F}'\mathbf{F})^{-1}\mathbf{F}'\mathbf{U}{-r}\mathbf{U}'{-r}\mathbf{F}| &= |\mathbf{I} - \mathbf{H}_2\mathbf{H}_3^{-1}| \ &= O_p\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) + O_p\left(\frac{1}{N^{\alpha}}\right) + O_p\left(\frac{N}{N^{\alpha}T}\right). \end{split}$$

The proof for (A.22) is almost identical. Using that V V ′ = I^N , we have

$$\begin{aligned} & \mathbf{\Lambda}'\mathbf{\Lambda} = \mathbf{\Lambda}'VV'\mathbf{\Lambda} \ & = \mathbf{\Lambda}'V_rV'r\mathbf{\Lambda} + \mathbf{\Lambda}'V{-r}V'_{-r}\mathbf{\Lambda} \ & = \mathbf{\Lambda}'\tilde{\mathbf{\Lambda}} \mathbf{D}r^{-2}\tilde{\mathbf{\Lambda}}'\mathbf{\Lambda}/N + \mathbf{\Lambda}'V{-r}V'_{-r}\mathbf{\Lambda} \ & = H_4H1^{-1}\mathbf{\Lambda}'\mathbf{\Lambda} + \mathbf{\Lambda}'V{-r}V'_{-r}\mathbf{\Lambda}, \end{aligned}$$

where H¹ = (Λ ′Λ)(Λ˜ ′ Λ) −1 , and H⁴ = (Λ ′Λ˜ /N)D⁻² r . Multiplying from the right by (Λ ′Λ) ⁻¹ and rearranging, we obtain

$$\begin{split} |(\mathbf{\Lambda}'\mathbf{\Lambda})^{-1}\mathbf{\Lambda}'\mathbf{V}{-r}\mathbf{V}'{-r}\mathbf{\Lambda}| &= |\mathbf{I} - \mathbf{H}_4\mathbf{H}_1^{-1}| \ &= O_p\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) + O_p\left(\frac{1}{N^{\alpha}}\right) + O_p\left(\frac{N}{N^{\alpha}T}\right), \end{split}$$

where the last equality again follows from Lemma 3 of Bai and Ng (2023). ■

Lemma 7

$$\frac{1}{N^{\alpha/2}} | (\mathbf{F}' \mathbf{F})^{-1} \mathbf{F}' \mathbf{e} \mathbf{V}_r | = O_p \left( \frac{1}{\sqrt{N^{\alpha} T}} \right) + O_p \left( \frac{1}{N^{\alpha}} \right) + O_p \left( \frac{N}{N^{\alpha}} \frac{1}{T} \right), \tag{A.23}$$

$$\frac{1}{N^{\alpha/2}} | \boldsymbol{e}_t' \boldsymbol{V}_r | = O_p \left( \sqrt{\frac{N}{N^{\alpha}}} \left[ \frac{N}{N^{\alpha}} \frac{1}{T} + \frac{1}{N^{\alpha}} + \frac{1}{\sqrt{N}} \right] \right). \tag{A.24}$$

Proof. We first prove (A.23). Using that Λ˜ = √ NV ^rDr, q N ^N^αD^r = Op(1), and the fact that Λ˜ = ^T ^X′F˜ , we have

$$\frac{1}{N^{\alpha/2}} | (\mathbf{F}'\mathbf{F})^{-1} \mathbf{F}' e \mathbf{V}_r | = O_p \left( \frac{1}{N^{\alpha}} \right) | (\mathbf{F}'\mathbf{F})^{-1} \mathbf{F}' e \tilde{\mathbf{\Lambda}} | = \frac{1}{TN^{\alpha}} | (\mathbf{F}'\mathbf{F})^{-1} \mathbf{F}' e e' \tilde{\mathbf{F}} | + \frac{1}{TN^{\alpha}} | (\mathbf{F}'\mathbf{F})^{-1} \mathbf{F}' e \mathbf{\Lambda} \mathbf{F}' \tilde{\mathbf{F}} | \leq | (\mathbf{F}'\mathbf{F}/T)^{-1} | | \tilde{\mathbf{F}}/\sqrt{T} | | \mathbf{F}/\sqrt{T} | \frac{1}{TN^{\alpha}} | e e' |

| (\mathbf{F}'\mathbf{F}/T)^{-1} | \frac{1}{N^{\alpha}T} | \mathbf{F}' e \mathbf{\Lambda} | | \mathbf{F}' \tilde{\mathbf{F}}/T | = O_p \left( \frac{1}{N^{\alpha}} \right) + O_p \left( \frac{N}{N^{\alpha}} \frac{1}{T} \right) + O_p \left( \frac{1}{\sqrt{N^{\alpha}T}} \right).$$ (A.25)

using that ∥ee′∥ = Op(max(N, T)) by Assumption 1, and ∥F ′ eΛ∥ = Op( N^αT) by As sumption 2(iii).

We now consider (A.24).

$$\frac{1}{N^{\alpha/2}} | \boldsymbol{e}{t}' \boldsymbol{V}{r} | = O{p} \left( \frac{1}{N^{\alpha}} \right) | \boldsymbol{e}{t}' \tilde{\boldsymbol{\Lambda}} | = \frac{1}{TN^{\alpha}} | \boldsymbol{e}{t}' \boldsymbol{e}' \tilde{\boldsymbol{F}} | + \frac{1}{TN^{\alpha}} | \boldsymbol{e}{t}' \boldsymbol{\Lambda} \boldsymbol{F}' \tilde{\boldsymbol{F}} | = \frac{1}{TN^{\alpha}} | \boldsymbol{e}{t}' \boldsymbol{e}' \boldsymbol{F} \boldsymbol{H}{0} | + \frac{1}{TN^{\alpha}} | \boldsymbol{e}{t}' \boldsymbol{e}' | | \tilde{\boldsymbol{F}} - \boldsymbol{F} \boldsymbol{H}{0} | + \frac{1}{N^{\alpha}} | \boldsymbol{e}{t}' \boldsymbol{\Lambda} | = O{p} \left( \frac{N}{N^{\alpha}} \delta{NT}^{-2} \right) + O{p} \left( \frac{N}{N^{\alpha}} \delta{NT}^{-1} \left( \frac{1}{N^{\alpha/2}} + \frac{1}{T} \frac{N}{N^{\alpha}} \right) \right) + O{p} \left( \frac{1}{\sqrt{N^{\alpha}}} \right).$$ (A.26)

where the first term and final term follow from Corollary 1 (ii), and the second term follows from Corollary 1 (i) and Proposition 1 in Bai and Ng (2023).

Lemma 8

$$\frac{1}{\sqrt{T}} | \boldsymbol{U}_r \boldsymbol{\varepsilon} | = O_p \left( \frac{1}{T^{1/2}} \right). \tag{A.27}$$

Proof.

$$\frac{1}{\sqrt{T}} | \boldsymbol{U}_r \boldsymbol{\varepsilon} | = \frac{1}{T} | \tilde{\boldsymbol{F}}' \boldsymbol{\varepsilon} | = O_p \left( \frac{1}{T^{1/2}} \right). \tag{A.28}$$

where last line uses Assumption 4 and Proposition 1 in Bai and Ng (2023).

Lemma 9 Under Assumption 1 –Assumption 5,

$$\sqrt{N^{\alpha}}(\tilde{\boldsymbol{f}}{T} - \boldsymbol{H}{4}^{\prime}\boldsymbol{f}{T}) = O{p}(1) + O{p}\left(\sqrt{\frac{N}{N^{\alpha}}}\frac{1}{T}\right) + O{p}\left(\frac{1}{N^{\alpha}}\right). \tag{A.29}$$

Proof : Bai and Ng (2023) Proposition 2 shows

$$\frac{1}{N} |\tilde{\boldsymbol{\Lambda}} - \boldsymbol{\Lambda} (\boldsymbol{H}_0')^{-1}|^2 = O_p \left(\frac{1}{T}\right) + O_p \left(\frac{1}{N^{1+\alpha}}\right), \tag{A.30}$$

$$\frac{1}{N^{\alpha}} |\tilde{\mathbf{\Lambda}} - \mathbf{\Lambda} (\mathbf{H}_0')^{-1}|^2 = O_p \left(\frac{N}{N^{\alpha}} \frac{1}{T}\right) + O_p \left(\frac{1}{N^{2\alpha}}\right). \tag{A.31}$$

Under Assumption 5, we then have

$$\frac{1}{N^{\alpha/2}} | (\tilde{\mathbf{\Lambda}} - \mathbf{\Lambda} (\mathbf{H}_4')^{-1})' \mathbf{e}_T | = O_p \left( \sqrt{\frac{N}{N^{\alpha}}} \frac{1}{T} \right) + O_p \left( \frac{1}{N^{\alpha}} \right). \tag{A.32}$$

this implies the result.

A.4 Preliminary results: random projection regression

Lemma 10 1. Define M = XER[R(R′X′XR) ⁻¹R′ ]X′ . Then, M = UDU′ where [D]ii = 1 + O^p ^N ^N^α k −1 for i = 1, . . . , r and [D]ii = Op(kδ⁻² NT ) for i = r + 1, . . . , δ² NT .

Define $$\mathbf{D}2 = \mathbb{E}[\mathbf{R}(\mathbf{R}'\mathbf{D}'{NT}\mathbf{D}{NT}\mathbf{R})^{-1}\mathbf{R}']\mathbf{D}'{NT}$$ . Then, $[\mathbf{D}2]{ii} = O_p\left(\sqrt{\frac{N}{N^{\alpha}}}\right)$ for $i = 1, \ldots, r$ and $[\mathbf{D}2]{ii} = Op(k\delta{NT}^{-1})$ for $i = r + 1, \ldots, \delta_{NT}^2$ .

Proof. We only prove part (i), part (ii) follows analogously. Recall that X/ √ NT = UDNTV ′ by (3). Using Lemma 3.

$$M = X \mathbb{E}_{R}[R(R'X'XR)^{-1}R']X'$$

$$= UD{NT}\mathbb{E}{R}[R(R'D'{NT}D{NT}R)^{-1}R']D'_{NT}U'$$

$$= UDU'.$$ (A.33)

By Lemma 5, the fact that the first r eigenvalues of X′X/NT are Op(N^α/N) and the next δ 2 NT −r eigenvalues are Op(δ −2 NT ) as established in Ahn and Horenstein (2013, Lemma A.9), D is a diagonal matrix with [D]ii = 1+O^p ^N ^N^α k −1 for i = 1, . . . , r and [D]ii = Op(k · δ −2 NT ) for i = r + 1, . . . , δ² NT .

Appendix B Proofs of the main results

B.1 Proof of Theorem 1 and Theorem 3

We have

$$\hat{y}{T+h|T}^{pca} = \tilde{\boldsymbol{f}}{T}(\tilde{\boldsymbol{F}}'\tilde{\boldsymbol{F}})^{-1}\tilde{\boldsymbol{F}}'\boldsymbol{y} \ = \tilde{\boldsymbol{f}}{T}'(\tilde{\boldsymbol{F}}'\tilde{\boldsymbol{F}})^{-1}\tilde{\boldsymbol{F}}'\boldsymbol{F}\boldsymbol{\gamma} + \tilde{\boldsymbol{f}}{T}'(\tilde{\boldsymbol{F}}'\tilde{\boldsymbol{F}})^{-1}\tilde{\boldsymbol{F}}'\boldsymbol{\varepsilon} \ = \boldsymbol{f}{T}'\boldsymbol{H}{0}(\tilde{\boldsymbol{F}}'\tilde{\boldsymbol{F}})^{-1}\tilde{\boldsymbol{F}}'\cdot\boldsymbol{F}\boldsymbol{\gamma} + (\tilde{\boldsymbol{f}}{T}' - \boldsymbol{f}{T}'\boldsymbol{H}{0})(\tilde{\boldsymbol{F}}'\tilde{\boldsymbol{F}})^{-1}\tilde{\boldsymbol{F}}'\boldsymbol{F}\boldsymbol{\gamma} + \tilde{\boldsymbol{f}}{T}'(\tilde{\boldsymbol{F}}'\tilde{\boldsymbol{F}})^{-1}\tilde{\boldsymbol{F}}'\boldsymbol{\varepsilon} \ = \boldsymbol{f}{T}'\boldsymbol{H}{0}\boldsymbol{H}{3}^{-1}\boldsymbol{\gamma} + (\tilde{\boldsymbol{f}}{T}' - \boldsymbol{f}{T}'\boldsymbol{H}{0})(\tilde{\boldsymbol{F}}'\tilde{\boldsymbol{F}})^{-1}\tilde{\boldsymbol{F}}'\boldsymbol{F}\boldsymbol{\gamma} + \tilde{\boldsymbol{f}}{T}'(\tilde{\boldsymbol{F}}'\tilde{\boldsymbol{F}})^{-1}\tilde{\boldsymbol{F}}'\boldsymbol{\varepsilon} \ = \boldsymbol{f}{T}'\boldsymbol{\gamma} + O{p}\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) + O{p}\left(\frac{1}{N^{\alpha}}\right) + O{p}\left(\frac{N}{N^{\alpha}T}\right) + |\tilde{\boldsymbol{f}}{T}' - \boldsymbol{f}{T}'\boldsymbol{H}{0}|O{p}(1) + O{p}\left(\frac{1}{\sqrt{T}}\right) \ = \boldsymbol{f}{T}'\boldsymbol{\gamma} + O{p}\left(\frac{1}{N^{\alpha/2}}\right) + O{p}\left(\frac{N^{3/2}}{N^{3\alpha/2}}\frac{1}{T}\right) + O{p}\left(\frac{N^{1/2}}{N^{3\alpha/2}}\right) + O_{p}\left(\frac{N^{2}}{N^{2\alpha}T^{1/2}}\right) \

O{p}\left(\frac{N^{2}}{N^{2\alpha}T^{3/2}}\right) + O{p}\left(\frac{1}{\sqrt{T}}\right), \tag{B.1}$$

where we use that H0H⁻¹ ³ = I +O^p √ 1 NαT +O^p 1 N^α +O^p ^N NαT by Lemma 3 in Bai and Ng (2023), and ∥f˜ ′ ^T − f ′ ^TH0∥ = O^p 1 Nα/² + O^p N3/² N3α/² 1 T + O^p N1/² N3α/² + O^p N² N2αT1/² + O^p N² N2αT3/² . To obtain the latter result, Bai and Ng (2023, p. 1901) shows that

$$\sqrt{N^{\alpha}}(\tilde{\boldsymbol{f}}{T}' - \boldsymbol{f}{T}'\boldsymbol{H}{0}) = \left(\frac{\tilde{\boldsymbol{\Lambda}}'\tilde{\boldsymbol{\Lambda}}}{N^{\alpha}}\right)^{-1}\boldsymbol{H}{4}^{-1}\sum{i=1}^{N}\boldsymbol{\Lambda}{i}e{iT} + \left(\frac{\tilde{\boldsymbol{\Lambda}}'\tilde{\boldsymbol{\Lambda}}}{N^{\alpha}}\right)^{-1}\frac{(\tilde{\boldsymbol{\Lambda}} - \boldsymbol{\Lambda}(\boldsymbol{H}{4}')^{-1})'\boldsymbol{e}_{T}}{\sqrt{N^{\alpha}}}.$$ (B.2)

The term on the right-hand side is Op(1) by Corollary 1 part (ii). The second term is covered by Bai and Ng (2023, Lemma A.2). This yields the result in (B.1) and completes the proof for Theorem 1.

Under Assumption 5, the idiosyncratic errors are independent over time. In that case, we can establish a much faster rate on the second term on the right-hand side of (B.2). From our construction of the PCA-based forecast, Λ˜ is estimated on data up to and including time t − h, and hence, independent of e^t . Then,

$$\mathbb{E}\left[\left(\frac{(\tilde{\mathbf{\Lambda}} - \mathbf{\Lambda}(\mathbf{H}{4}')^{-1})'\mathbf{e}{T}}{\sqrt{N^{\alpha}}}\right)^{2}\right] = \frac{1}{N^{\alpha}}\mathbb{E}\left[(\tilde{\mathbf{\Lambda}} - \mathbf{\Lambda}(\mathbf{H}{4}')^{-1})'\mathbf{G}{N}(\tilde{\mathbf{\Lambda}} - \mathbf{\Lambda}(\mathbf{H}_{4}')^{-1})\right]$$

$$\leq \frac{C}{N^{\alpha}}\mathbb{E}\left[(\tilde{\mathbf{\Lambda}} - \mathbf{\Lambda}(\mathbf{H}{4}')^{-1})'(\tilde{\mathbf{\Lambda}} - \mathbf{\Lambda}(\mathbf{H}{4}')^{-1})\right]$$

$$= O\left(\frac{N}{N^{\alpha}}\frac{1}{T}\right) + O\left(\frac{1}{N^{2\alpha}}\right).$$ (B.3)

Using Assumption 3, we then have

$$|\tilde{\boldsymbol{f}}_T' - \boldsymbol{f}_T' \boldsymbol{H}_0| = O_p \left(\frac{1}{N^{\alpha/2}}\right). \tag{B.4}$$

This completes the proof of Theorem 3.

B.2 Proof of Theorem 2 and Theorem 4

Using (1) and the fact that (F ′F) ⁻¹F ′X = Λ ′ + (F ′F) ⁻¹F ′ e, we can rewrite the forecast (9) as

$$\begin{split} \hat{y}{T+h|T}^{rp} &= \mathbf{f}{T}' \mathbf{\Lambda}' \mathbb{E}{R} [\mathbf{R} (\mathbf{R}' \mathbf{X}' \mathbf{X} \mathbf{R})^{-1} \mathbf{R}'] \mathbf{X}' \mathbf{F} \boldsymbol{\gamma} + \mathbf{e}{T}' \mathbb{E}{R} [\mathbf{R} (\mathbf{R}' \mathbf{X}' \mathbf{X} \mathbf{R})^{-1} \mathbf{R}'] \mathbf{X}' \mathbf{F} \boldsymbol{\gamma} \ &+ \mathbf{x}{T}' \mathbb{E}{R} [\mathbf{R} (\mathbf{R}' \mathbf{X}' \mathbf{X} \mathbf{R})^{-1} \mathbf{R}'] \mathbf{X}' \boldsymbol{\varepsilon} \ &= \mathbf{f}{T}' \boldsymbol{\gamma} + \underbrace{\mathbf{f}{T}' [(\mathbf{F}' \mathbf{F})^{-1} \mathbf{F}' \mathbf{M} \mathbf{F} - \mathbf{I}] \boldsymbol{\gamma}}{(I)} - \underbrace{\mathbf{f}{T}' (\mathbf{F}' \mathbf{F})^{-1} \mathbf{F}' \mathbf{e} \mathbf{M}{2} \mathbf{F} \boldsymbol{\gamma}}{(II)} + \underbrace{\mathbf{e}{T}' \mathbf{M}{2} \mathbf{F} \boldsymbol{\gamma}}{(III)} - \underbrace{\mathbf{b}{T}' \mathbf{M} \boldsymbol{\varepsilon}}{(IV)}. \end{split}$$

$$(B.5)$$

where M = XER[R(R′X′XR) ⁻¹R′ ]X′ = UDU′ . Lemma 10 shows that [D]ii = 1 + O^p ^N N^α k for i = 1, . . . , r and [D]ii = O^p kδ⁻² NT for i > r. We also define

$$\boldsymbol{M}_2 = \mathbb{E}_R[\boldsymbol{R}(\boldsymbol{R}'\boldsymbol{X}'\boldsymbol{X}\boldsymbol{R})^{-1}\boldsymbol{R}']\boldsymbol{X}' = \frac{1}{\sqrt{NT}}\boldsymbol{V}\boldsymbol{D}_2\boldsymbol{U}',$$

with [D2]ii = O^p ^N N^α for i = 1, . . . , r and [D]ii = O^p kδ⁻¹ NT for i > r. b^T is a T × 1 vector with the Tth element equal to one and the only nonzero element. The expansion (B.5) also holds for the ridge forecast (7).

We now bound the terms (I)−(IV ). The difference between Theorem 2 and Theorem 4 is in the analysis of term (III).

Term (I) Assuming that kδ⁻² NT → 0 and ^N ^N^α k ⁻¹ → 0,

$$\begin{split} |(I)| &= |\boldsymbol{f}{T}'| |(\boldsymbol{F}'\boldsymbol{F})^{-1}\boldsymbol{F}'\boldsymbol{U}\boldsymbol{D}\boldsymbol{U}'\boldsymbol{F} - \boldsymbol{I}| |\boldsymbol{\gamma}| \ &\leq |\boldsymbol{f}{T}'| |(\boldsymbol{F}'\boldsymbol{F})^{-1}\boldsymbol{F}'\boldsymbol{U}{r}\boldsymbol{U}{r}'\boldsymbol{F} - \boldsymbol{I}| |\boldsymbol{\gamma}| + O{p}\left(\frac{N}{N^{\alpha}}k^{-1}\right) |\boldsymbol{f}{T}'| |(\boldsymbol{F}'\boldsymbol{F}/T)^{-1}| \frac{1}{T}|\boldsymbol{F}'\boldsymbol{U}{r}|^{2} |\boldsymbol{\gamma}| \ &+ O{p}(k\cdot\delta{NT}^{-2})|\boldsymbol{f}{T}'| |(\boldsymbol{F}'\boldsymbol{F})^{-1/2}\boldsymbol{F}'\boldsymbol{U}{-r}|^{2} |\boldsymbol{\gamma}| \ &= O{p}\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) + O{p}\left(\frac{1}{N^{\alpha}}\right) + O{p}\left(\frac{N}{N^{\alpha}T}\right) + O_{p}\left(\frac{N}{N^{\alpha}}k^{-1}\right). \end{split}$$

To obtain the final line, we applied (A.20) to the first and second term after the inequality sign, and we applied (A.21) to the third term after the inequality sign.

Term (II) We have,

$$\begin{split} |(II)| &= |\boldsymbol{f}_T'(\boldsymbol{F}'\boldsymbol{F}/T)^{-1}\frac{1}{T}\boldsymbol{F}'\boldsymbol{e}\frac{1}{\sqrt{NT}}\boldsymbol{V}\boldsymbol{D}_2\boldsymbol{U}'\boldsymbol{F}\boldsymbol{\gamma}| \ &\leq |\boldsymbol{f}_T||\boldsymbol{\gamma}|\frac{1}{\sqrt{N^{\alpha}}}|(\boldsymbol{F}'\boldsymbol{F})^{-1}\boldsymbol{F}'\boldsymbol{e}\boldsymbol{V}_r|\frac{1}{\sqrt{T}}|\boldsymbol{U}_r\boldsymbol{F}|O_p(1) + \ &Op(k\delta{NT}^{-1})|\boldsymbol{f}T||\boldsymbol{\gamma}|\frac{1}{\sqrt{NT}}|\boldsymbol{F}'\boldsymbol{e}|\frac{1}{\sqrt{T}}|(\boldsymbol{F}'\boldsymbol{F}/T)^{1/2}||(\boldsymbol{F}'\boldsymbol{F})^{-1/2}\boldsymbol{F}'\boldsymbol{U}{-r}| \ &= O_p\left(\frac{1}{N^{\alpha}}\right) + O_p\left(\frac{N}{N^{\alpha}}\frac{1}{T}\right) + O_p\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) \ &+ Op\left(k\delta{NT}^{-1}\frac{1}{\sqrt{T}}\left[\frac{1}{(N^{\alpha}T)^{1/4}} + \frac{1}{N^{\alpha/2}} + \sqrt{\frac{N}{N^{\alpha}}\frac{1}{T}}\right]\right), \end{split}$$

with the first three terms following from (A.23), and the final terms due to (A.21) and the fact that ¹ NT ∥F ′ e∥ ² = Op(1) by Lemma A.1 in Bai and Ng (2023).

Term (III): Theorem 2, correlation in et.

$$\begin{aligned} |\boldsymbol{e}{T}'\boldsymbol{M}{2}\boldsymbol{F}\boldsymbol{\gamma}| &= \frac{1}{\sqrt{NT}} |\boldsymbol{e}{T}'\boldsymbol{V}\boldsymbol{D}{2}\boldsymbol{U}'\boldsymbol{F}\boldsymbol{\gamma}| \ &= \frac{1}{\sqrt{N}} O{p} \left( \sqrt{\frac{N}{N^{\alpha}}} \right) |\boldsymbol{e}{T}'\boldsymbol{V}{r}| |\frac{1}{\sqrt{T}} \boldsymbol{U}{r}'\boldsymbol{F}\boldsymbol{\gamma}| \ &+ \frac{1}{\sqrt{N}} O{p}(k\delta{NT}^{-1}) |\boldsymbol{e}{T}'\boldsymbol{V}{-r}| |\frac{1}{\sqrt{T}} \boldsymbol{U}{-r}'\boldsymbol{F}\boldsymbol{\gamma}| \ &= O{p} \left( \frac{1}{N^{\alpha/2}} \right) \ &+ O{p} \left( \frac{N^{3/2}}{N^{3\alpha/2}} \frac{1}{T} \right) + O{p} \left( \frac{N^{1/2}}{N^{3\alpha/2}} \right) + O{p} \left( \frac{N^{2}}{N^{2\alpha}T^{1/2}} \right) + O{p} \left( \frac{N^{2}}{N^{2\alpha}T^{3/2}} \right) \ &+ O{p} \left( k\delta{NT}^{-1} \left[ \frac{1}{(N^{\alpha}T)^{1/4}} + \frac{1}{N^{\alpha/2}} + \sqrt{\frac{N}{N^{\alpha}}} \frac{1}{T} \right] \right), \end{aligned} \tag{B.6}$$

where we used (A.24) to obtain the terms on the second to last line, and (A.21) and the fact that N ⁻1/²∥e ′ ^TV ⁻r∥ ≤ N ⁻1/²∥e^T ∥ = Op(1) by Assumption 1 to obtain the final line.

Term (III): Theorem 4, serial correlation in et. If e^t is independent with {Fs, es}s<t then we can derive a tighter bound. In this case, term (III) is mean zero. We then bound

$$\operatorname{Var}\left(\frac{1}{\sqrt{NT}}\boldsymbol{e}{T}'\boldsymbol{V}{r}\boldsymbol{D}{2,r}\boldsymbol{U}{r}'\boldsymbol{F}\boldsymbol{\gamma}\right) \leq C\frac{\lambda{\max}(\mathbb{E}[\boldsymbol{e}{T}'\boldsymbol{e}_{T}])}{N^{\alpha}},$$

$$\operatorname{Var}\left(\frac{1}{\sqrt{NT}}\boldsymbol{e}{T}'\boldsymbol{V}{-r}\boldsymbol{D}{2,-r}\boldsymbol{U}{-r}'\boldsymbol{F}\boldsymbol{\gamma}\right) \leq \frac{1}{N}k^{2}\delta{NT}^{-2}\left(O{p}\left(\frac{1}{\sqrt{N^{\alpha}T}}\right) + O{p}\left(\frac{1}{N^{\alpha}}\right) + O{p}\left(\frac{N}{N^{\alpha}T}\right)\right).$$ (B.7)

We can then improve the bound on (III) as follows.

$$|\boldsymbol{e}{T}'\boldsymbol{M}{2}\boldsymbol{F}\boldsymbol{\gamma}| \leq \frac{1}{\sqrt{NT}}|\boldsymbol{e}{T}'\boldsymbol{V}{r}\boldsymbol{D}{2,r}\boldsymbol{U}{r}'\boldsymbol{F}\boldsymbol{\gamma}| + \frac{1}{\sqrt{NT}}|\boldsymbol{e}{T}'\boldsymbol{V}{-r}\boldsymbol{D}{2,-r}\boldsymbol{U}{-r}'\boldsymbol{F}\boldsymbol{\gamma}|$$

$$= O{p}\left(\frac{1}{N^{\alpha/2}}\right) + O{p}\left(k\delta_{NT}^{-1}\frac{1}{\sqrt{N}}\left[\frac{1}{(N^{\alpha}T)^{1/4}} + \frac{1}{N^{\alpha/2}} + \sqrt{\frac{N}{N^{\alpha}}\frac{1}{T}}\right]\right). \tag{B.8}$$

Term (IV ) By Assumption 4, we have b ′ ^TM ε/ p b TM² b^T = Op(1). Then,

$$||(IV)|| = \sqrt{\mathbf{b}_T' \mathbf{M}^2 \mathbf{b}_T} \frac{\mathbf{b}_T' \mathbf{M} \varepsilon}{\sqrt{\mathbf{b}_T' \mathbf{M}^2 \mathbf{b}_T}}$$

$$= O_p \left(\frac{1}{\sqrt{T}}\right) + Op \left(k \delta{NT}^{-2}\right).$$ (B.9)

Combining the established bounds yields Theorem 2 and Theorem 4.

Appendix C FRED-MD and FRED-QD subset

Table 1 presents the list of variables that are both in the FRED-MD and the FRED-QD, with the id as listed in the FRED-MD and the FRED-QD. The column group corresponds to the groups of the FRED-MD: (1) Output and income; (2) Labor market; (3) Housing; (4) Consumption, orders, and inventories; (5) Money and credit; (6) Interest and exchange rates; (7) Prices; (8) Stock market.

The column tcode is the transformation applied to the raw data x, as specified in the FRED-QD: (1) no transformation; (2) ∆x^t ; (3) ∆²x^t ; (4) log(xt); (5) ∆ log(xt); (6) ∆² log(xt). The transformation code differs between the FRED-MD and FRED-QD for 15 variables. For example, Housing Starts series are suggested to be transformed by taking logs in the FRED-MD, but the recommendation in the FRED-QD is to apply log differences. In those cases, we follow the FRED-QD.

Table 1: Variables in FRED-MD and FRED-QD

id (M)	id (Q)	Group (M)	tcode (Q)	Mnemonic (M)	Description
6	22	1	5	INDPRO	IP Index
8	23	1	5	IPFINAL	IP: Final Products and Nonindustrial Supplies
9	24	1	5	IPCONGD	IP: Final Products (Market Group)
10	28	1	5	IPDCONGD	IP: Consumer Goods
11	30	1	5	IPNCONGD	IP: Durable Consumer Goods
12	31	1	5	IPBUSEQ	IP: Nondurable Consumer Goods
13	25	1	5	IPMAT	IP: Materials
14	26	1	5	IPDMAT	IP: Durable Materials
15	27	1	5	IPNMAT	IP: Nondurable Materials
16	194	1	5	IPMANSICS	IP: Manufacturing (SIC)
17	195	1	5	IPB51222S	IP: Residential Utilities
18	196	1	5	IPFUELS	IP: Fuels
19	34	1	1	CUMFNS	Capacity Utilization: Manufacturing
20	80	2	1	HWI	Help-Wanted Index for United States
21	220	2	2	HWIURATIO	Ratio of Help Wanted/No. Unemployed
23	57	2	5	CE16OV	Civilian Employment
24	59	2	2	UNRATE	Civilian Unemployment Rate
25	197	2	2	UEMPMEAN	Average Duration of Unemployment (Weeks)
26	65	2	5	UEMPLT5	Civilians Unemployed - Less Than 5 Weeks
27	66	2	5	UEMP5TO14	Civilians Unemployed for 5-14 Weeks
29	67	2	5	UEMP15T26	Civilians Unemployed for 15-26 Weeks
30	68	2	5	UEMP27OV	Civilians Unemployed for 27 Weeks and Over
31	221	2	5	CLAIMSx	Initial Claims
32	35	2	5	PAYEMS	All Employees: Total nonfarm
33	39	2	5	USGOOD	All Employees: Goods-Producing Industries
35	42	2	5	USCONS	All Employees: Construction
36	37	2	5	MANEMP	All Employees: Manufacturing
	40	2	5	DMANEMP	All Employees: Durable goods
37	41	2	5	NDMANEMP	All Employees: Nondurable goods
38	38	2	5	SRVPRD	All Employees: Service-Providing Industries
39	50	2	5	USTPU	All Employees: Trade, Transportation & Utilities
40
41	53	2	5	USWTRADE	All Employees: Wholesale Trade
42	52	2	5	USTRADE	All Employees: Retail Trade
43	44	2	5	USFIRE	All Employees: Financial Activities
44	51	2	5	USGOVT	All Employees: Government
45	198	2	2	CES0600000007	Avg Weekly Hours : Goods-Producing
46	79	2	2	AWOTMAN	Avg Weekly Overtime Hours : Manufacturing
47	77	2	1	AWHMAN	Avg Weekly Hours : Manufacturing
120	216	2	6	CES0600000008	Avg Hourly Earnings : Goods-Producing
48	81	3	5	HOUST	Housing Starts: Total New Privately Owned
49	85	3	5	HOUSTNE	Housing Starts, Northeast
50	84	3	5	HOUSTMW	Housing Starts, Midwest
51	86	3	5	HOUSTS	Housing Starts, South
52	87	3	5	HOUSTW	Housing Starts, West
53	83	3	5	PERMIT	New Private Housing Permits (SAAR)
54	227	3	5	PERMITNE	New Private Housing Permits, Northeast (SAAR)
55	228	3	5	PERMITMW	New Private Housing Permits, Midwest (SAAR)
56	229	3	5	PERMITS	New Private Housing Permits, South (SAAR)
57	230	3	5	PERMITW	New Private Housing Permits, West (SAAR)
59	90	4	5	AMDMNOx	New Orders for Durable Goods
60	93	4	5	ANDENOx	New Orders for Nondefense Capital Goods

Table 1: Variables in FRED-MD and FRED-QD (continued)

id (M)	id (Q)	Group (M)	tcode (Q)	Mnemonic (M)	Description
61	92	4	5	AMDMUOx	Unfilled Orders for Durable Goods
62	222	4	5	BUSINVx	Total Business Inventories
63	223	4	2	ISRATIOx	Total Business: Inventories to Sales Ratio
123	188	4	1	UMCSENTx	Consumer Sentiment Index
66	161	5	5	M2REAL	Real M2 Money Stock
68	199	5	6	TOTRESNS	Total Reserves of Depository Institutions
69	200	5	7	NONBORRES	Reserves Of Depository Institutions
73	224	5	2	CONSPI	Nonrevolving consumer credit to Personal Income
124	217	5	6	DTCOLNVHFNM	Consumer Motor Vehicle Loans Outstanding
125	218	5	6	DTCTHFNM	Total Consumer Loans and Leases Outstanding
126	219	5	6	INVEST	Securities in Bank Credit at All Commercial Banks
78	144	6	2	FEDFUNDS	Effective Federal Funds Rate
79	225	6	2	CP3Mx	3-Month AA Financial Commercial Paper Rate
80	145	6	2	TB3MS	3-Month Treasury Bill
81	146	6	2	TB6MS	6-Month Treasury Bill
82	147	6	2	GS1	1-Year Treasury Rate
83	201	6	2	GS5	5-Year Treasury Rate
84	148	6	2	GS10	10-Year Treasury Rate
85	150	6	2	AAA	Moody's Seasoned Aaa Corporate Bond Yield
86	151	6	2	BAA	Moody's Seasoned Baa Corporate Bond Yield
87	226	6	1	COMPAPFFx	3-Month Commercial Paper Minus FEDFUNDS
88	202	6	1	TB3SMFFM	3-Month Treasury C Minus FEDFUNDS
91	203	6	1	T5YFFM	5-Year Treasury C Minus FEDFUNDS
93	204	6	1	AAAFFM	Moody's Aaa Corporate Bond Minus FEDFUNDS
95	182	6	5	TWEXAFEGSMTHx	Trade Weighted U.S. Dollar Index
96	184	6	5	EXSZUSx	Switzerland / U.S. Foreign Exchange Rate
97	185	6	5	EXJPUSx	Japan / U.S. Foreign Exchange Rate
98	186	6	5	EXUSUKx	U.S. / U.K. Foreign Exchange Rate
99	187	6	5	EXCAUSx	Canada / U.S. Foreign Exchange Rate
100	122	7	6	WPSFD49207	PPI: Finished Goods
101	124	7	6	WPSFD49502	PPI: Finished Consumer Goods
102	127	7	6	WPSID61	PPI: Intermediate Materials
103	205	7	6	WPSID62	PPI: Crude Materials
104	130	7	5	OILPRICEx	Crude Oil, spliced WTI and Cushing
105	206	7	6	PPICMM	PPI: Metals and metal products
106	120	7	6	CPIAUCSL	CPI : All Items
107	207	7	6	CPIAPPSL	CPI : Apparel
108	208	7	6	CPITRNSL	CPI : Transportation
109	209	7	6	CPIMEDSL	CPI : Medical Care
110	210	7	6	CUSR0000SAC	CPI : Commodities
111	211	7	6	CUSR0000SAD	CPI : Durables
112	212	7	6	CUSR0000SAS	CPI : Services
113	213	7	6	CPIULFSL	CPI : All Items Less Food
114	214	7	6	CUSR0000SA0L2	CPI : All items less shelter
115	215	7	6	CUSR0000SA0L5	CPI : All items less medical care
116	95	7	6	PCEPI	Personal Cons. Expend.: Chain Index
74	246	8	5	S&P 500	S&P's Common Stock Price Index: Composite
75	245	8	5	S&P: indust	S&P's Common Stock Price Index: Industrials
76	247	8	2	S&P div yield	S&P's Composite Common Stock: Dividend Yield
77	248	8	5	S&P PE ratio	S&P's Composite Common Stock: Price-Earnings Ratio
127	178	8	1	VIXCLSx	VIX

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