Limit Theorems for Factor Models∗

Stanislav Anatolyev† Anna Mikusheva‡ CERGE-EI and NES MIT

September 2020

Abstract

The paper establishes central limit theorems and proposes how to perform valid inference in factor models. We consider a setting where many counties/regions/assets are observed for many time periods, and when estimation of a global parameter in- cludes aggregation of a cross-section of heterogeneous micro-parameters estimated separately for each entity. The central limit theorem applies for quantities involving both cross-sectional and time series aggregation, as well as for quadratic forms in time-aggregated errors. The paper studies the conditions when one can consistently estimate the asymptotic variance, and proposes a bootstrap scheme for cases when one cannot. A small simulation study illustrates performance of the asymptotic and bootstrap procedures. The results are useful for making inferences in two-step estimation procedures related to factor models, as well as in other related contexts. Our treatment avoids structural modeling of cross-sectional dependence but imposes time-series independence.

Keywords: factor models, two-step procedure, dimension asymptotics, central limit theorem.

JEL classiﬁcation codes: C13, C33, C38, C55.

∗We thank the Editor Peter C.B. Phillips, Co-Editor Guido Kuersteiner, and three anonymous referees for guidance and helpful suggestions that have greatly improved the paper. †Address: Stanislav Anatolyev, CERGE-EI, Politickych vˇezˇn˚u7, 11121 Prague 1, Czech Republic; e-mail [email protected]. Czech Science Foundation support under grants 17-26535S and 20-28055S is gratefully acknowledged. ‡Address: Department of Economics, M.I.T., 50 Memorial Drive, Building E52, Cambridge, MA, 02142, USA; e-mail [email protected]. National Science Foundation support under grant 1757199 is gratefully acknowledged. 1 INTRODUCTION

Data with an underlying factor structure are increasingly used in empirical macroeconomics and finance. Often these data consist of time series of observations for multiple cross-sectional units (assets, portfolios, regions or industries). Quite a few new estimation strategies have appeared in the empirical literature that use both cross-sectional and time series variation in order to estimate global structural parameters. Often the parameter of interest arises from aggregation or estimation using cross-sectional variation of individual parameters for each entity. One example of such a structure is linear factor pricing model in asset pricing (Fama & MacBeth 1973 and Shanken 1992), where for estimation we usually use time series of excess returns for a number of portfolios or assets priced by a small number of risk factors. Each portfolio or stock may have its own (heterogeneous) exposure to risk, often referred to as betas, which can be estimated separately from time series observations for each portfolio. The parameter of interest, a risk premium, is defined as the coefficient of proportionality in the cross-sectional relation between the average excess return on a portfolio and its individual beta. A vast majority of macroeconomic shocks are only weakly identified via structural VARs that use only time series observations on leading macro variables. A new approach to the estimation of causal effects of a macro shock on the economy is to use cross-sectional variation in data on regions, countries or industries. For example, Serrato & Wingender (2016) use cross-sectional variation in federal spending programs due to a Census shock to identify the causal impact of government spending on the economy. Cross-sectional variation among counties in government spending and in the accuracy of census-based estimates of population provides a better justified treatment effect framework, allows for the estimation of local fiscal multipliers, and finally gives a better global estimate of the fiscal multiplier via aggregation of local multipliers. Hagedorn et al. (2015) estimate the aggregate effect of unemployment-benefit duration on employment and labor force participation using cross-sectional differences across US states. Sarto (2018) discusses how heterogeneous sensitivities of regions to aggregate policy variables, so called micro- global elasticities, can be used to recover macro elasticities of interest such as, for example, a fiscal multiplier. A shared feature of the above-mentioned examples is the use of time-series observations on multiple entities (stocks, portfolios, counties, states or industries), while data on those entities are not independent and identically distributed. Moreover, variables for different entities often display strong co-movements to the extent that the data have

1 a factor structure, and estimation of these co-movements is the main goal. Indeed, the realization of a risk factor in the economy moves returns on all portfolios simultaneously, while a federal fiscal shock moves spending in all US counties, though in both cases het- erogeneously so. A valid estimation procedure must explicitly model and account for the data’s factor structure to the extent that the error terms (or residuals) can be considered idiosyncratic; see Kleibergen & Zhan (2015) and Anatolyev & Mikusheva (2018) for how a factor structure that is unaccounted for can lead to misleading results. However, idiosyncrasy of the errors usually implies only that the correlation among errors for different entities is relatively small and does not introduce first-order bias to the estimation procedure. Usually, it is not reasonable to assume that errors for different entities are completely independent; indeed, stocks in the same industry are likely to co-move even after global-economy risks are removed, while errors for neighboring counties are more likely to be correlated even after one accounts for federal shocks. At the same time, we typically want to remain agnostic about the correlation structure of shocks and avoid their structural modeling as long as this does not introduce biases. The second typical feature of the above-mentioned examples is the two-step nature of the estimation procedure, where in the first step we estimate entity-specific coefficients (risk exposures/betas, local fiscal multipliers, micro-global elasticities) by running a time- series regression separately for each entity. In the second step, we estimate the global coefficient of interest by either aggregating entity-specific coefficients (Serrato & Wingen- der 2016 and Hagedorn et al. 2015), or by running an OLS regression on the cross-section of entity-specific coefficients (Fama & MacBeth 1973 and Sarto 2018), or by running an IV regression on the cross-section of entity-specific coefficients (Anatolyev & Mikusheva 2018). The goal of this paper is to establish central limit theorems (CLTs) and to provide a tool for establishing asymptotic normality of estimates obtained in such two-step estimation procedures and for finding ways to do asymptotically correct inference, while being flexible in modeling the cross-sectional dependence of errors. The main difficulty here is that even though the second step cross-sectional regression has nearly uncorrelated errors (which is usually sufficient to obtain consistency of the two-step estimator), this condition is usually insufficient for a CLT, which typically requires that stronger discipline be imposed on the dependence structure (such as independence, or a martingale difference structure, or mixing). Our solution to this problem is to restrict the time series behavior while staying agnostic about the cross-sectional dependence. We assume time-series independence of idiosyncratic errors, which is consistent with market efficiency for factor

2 asset pricing models and the non-predictability of macro shocks in macroeconomic settings. The estimation noise in a two-stage procedure involves aggregation both over time (from the first step) and over entities (from the second step). We show that under certain conditions it is sufficient to have a CLT over just one of these directions, and we use the time-series direction for that. When the second step uses an OLS or IV estimator, the CLT must adapt to averages of quadratic forms, as both the second-step-dependent variable and the second stage regressor/instrument contain first-stage estimation noise. Our CLT has a linear and a quadratic part. We also note that a need for a CLT for quadratic forms in factor models sometimes arises for the first-step estimators (e.g., Pesaran & Yamagata 2018) or in higher order asymptotic derivations (e.g., Bai & Ng 2010). There is a growing literature that establishes different CLTs while acknowledging the importance of cross-sectional dependence in the data, which stems from spatial relations and/or from the presence of common factors. Kuersteiner & Prucha (2013) establish a CLT for linear sums in a panel data context with growing cross-sectional dimension N and fixed time-series dimension T allowing for cross-sectional dependence, and Kuersteiner & Prucha (2020) extend these results to quadratic forms as well. Both papers impose conditional moment restrictions, which allows the authors to construct a martingale difference sequence in the cross-sectional direction. The main conditional moment restrictions imply a correct specification of an underlying model, which need not be required by our CLT. However, the mentioned papers allow more flexibility in modeling the time dependence, and do not require large T . Another CLT that requires both large N and large T is established in Hahn et al. (2020) for linear terms only. This paper also contributes to the literature on the CLT for quadratic forms. Various types of CLTs for quadratic forms have been previously established and used in the many instrument literature (see for example, Chao et al. 2012, Hausman et al. 2012, Sølvsten 2020) and many covariate literature (see Cattaneo et al. 2018), as well as in the literature on semi-parametric estimation (Cattaneo et al. 2014a, 2014b). The CLT used in those papers are established for the cross-sectional dimension only, and rely heavily on the independence assumption. We adapt the ideas used in Chao et al. (2012), specifically the approach of de Jong (1987), to accommodate large cross-sectionally dependent panels; an alternative approach, known as Stein’s method, is used in Sølvsten (2020). Our second set of results is related to ways of conducting valid statistical inference. Under strengthened conditions on the weakness of the cross-sectional correlation of errors, we show that a conventional variance estimator is consistent, and so the usual asymptotic

3 inference can be applied. When such strengthened conditions do not hold, we propose instead a variant of a wild bootstrap scheme that replicates the original cross-sectional dependence structure. We also conduct a small simulation experiment that provides evidence on the approximation quality of our CLT and on the empirical size and power of wild bootstrap in a moderately sized panel. The paper proceeds as follows. Section 2 explains problems with establishing asymptotic Gaussianity for two-step and other estimators and test statistics, and shows how discipline in the time series direction can help. Section 3 introduces assumptions on idiosyncratic errors, states central limit theorems for two cases, and discusses the relevance of those cases to empirical practice. Section 4 discusses estimation of asymptotic variances for asymptotic inference and alternative inference tools based on the bootstrap. Section 5 presents a small simulation experiment that reveals properties of asymptotic and proposed bootstrap inference tools. Section 6 concludes. All proofs appear in the Appendix.

2 GOALS AND EXAMPLES

Let the data contain observations on many units indexed by i = 1, ..., N, and observed for multiple time periods t =1, ..., T. We assume that both N and T increase to infinity without restrictions on their rates. The goal of this paper is to find the conditions under which the following statement will hold: 1 N Ξ ξ (0, Σ ), (1) N,T ≡ √ i ⇒N ξ N i=1 X where 1 T v γ e √T s=1 s i is ξi = , 1 T w e e T s=1P t~~mean the factor-removed part of entity-specific variables.~~

4 As we argue below (see Examples 1–3), statements (1) and (2) are often needed in order to conduct statistical inferences (testing or conﬁdence set construction) about a structural parameter, λ, which is estimated in two steps. We consider a case when in the

ﬁrst step a researcher estimates a parameter βi for each entity/unit/state i = 1, ..., N, typically via running OLS or IV time series regressions. A typical linear estimator can be written as βi = βi + εi, where the estimation error has the structure εi = 1 + o (1) 1 T v e , with the o (1) term uniformly small over the units. In this setting, v p T t=1 t it p t b is either P a regressor common to all entities, or a common systematic part of entity-speciﬁc regressors that have a factor structure.2

Example 1. There is a variety of estimation approaches that can be used at the second step. The simplest of them is weighted averaging of the ﬁrst step estimates, viz. λ = 1 N N i=1 γiβi. Such an estimator is used in Sarto (2018). In order to justify asymptotic b Gaussianity of λ and to make statistical inferences about λ, one needs statements on the P b asymptotic behavior of b T N 1 N 1 T γ ε = v γ e . N i i √ √ t i it r i=1 N i=1 T t=1 X X X Note that the last expression has the structure of normalized averages stated as the ﬁrst

component of ΞN,T from equation (1). Such ‘linear’ terms, where only the ﬁrst component of ΞN,T is involved, are very common in asymptotic derivations in factor models (e.g., Bai & Ng 2006, 2010).

Example 2. The second estimation step may invoke a more complex estimator involving a sample covariance between multiple first stage estimators or estimators for multiple first stage parameters. For example, the Fama-MacBeth procedure employs the data on excess returns to a set of portfolios r , i =1, ..., N, t =1, ..., T and time series of a risk factor { it } F , t =1, ..., T . Namely, it uses two collections of first stage parameters – the average { t } (1) E (1) 1 T return on a portfolio βi = rit via the sample average return βi = T t=1 rit, and the 2 Our setting can accommodate entity-specific regressors, say vit, thatb have a factorP structure them- selves. Assume that vit = aiut + uit, where ut is a common co-movement in the regressors and uit is idiosyncratic. Then

~~T T T T 1 1 1 1 ε = v e = u (a e )+ u e = v e∗ , i T it it T t i it T it it T t it t=1 t=1 t=1 t=1 X X X X ′ ∗ where vt = (ut, 1) and eit = (aieit,uiteit).~~

5 (2) 1 risk exposure of a portfolio βi = var(Ft)− cov(rit, Ft) via the time series OLS regression (2) of rit on Ft resulting in an estimate βi . At the second stage of the Fama-MacBeth (1) procedure, one runs the OLS regression of the sample average return βi on the portfolio b (2) risk exposure estimated at the first step βi . In this case, b N N (2)b (2) 1 (1) (2) − λ = βi βi βi βi , i=1 i=1 X X b b b b b the second step involves two sample covariances. If one wants to derive the asymptotic distribution of λ, one needs to establish the asymptotic distribution for a properly normalized 1 N (1) (2) (j) (j) (j) sample covariance of the two first step estimators N i=1 βi βi , where βi = βi +εi , b (j) 1 T (j) with the estimation error having the structure εi = 1+ op(1) T t=1 vt eit, the term P b b b op(1) being uniform in i. P The normalized sample covariance of the two first step estimators contains several terms:

1 N β(1)β(2) E β(1)β(2) √ i i − i i N i=1 X b b b b 1 N 1 N = β(1)ε(2) + β(2)ε(1) + ε(1)ε(2) E ε(1)ε(2) . (3) √ i i i i √ i i − i i N i=1 N i=1 X X The first term on the right-hand-side of equation (3) is similar to a weighted average of (j) first step estimators and has the form of the first component of ΞN,T (treating βi as constants similar to constants γi). The second term in equation (3) is more complicated and calls for a Central Limit Theorem for quadratic forms:

~~T N ε(1)ε(2) E ε(1)ε(2) √ i i − i i N i=1 X 1 N T 1 N 1 T = v(1)v(2) e2 E[e2 ] + w e e , √ t t it − it √ T st it is T N i=1 t=1 N i=1 s=1 t~~

Example 3. Anatolyev & Mikusheva (2018) propose a split-sample estimator as an alternative to the Fama-MacBeth procedure for factor asset pricing. There are three sets (1) of parameter estimates produced at the ﬁrst stage: the sample average return βi =

b 6 1 T T t=1 rit and two estimates of the portfolio risk exposure computed as OLS estimates (2) (3) inP regressions of rit on Ft on diﬀerent sub-samples, say, βi and βi . In our notation, the usage of a sub-sample is accommodated by setting vt = 0 for those t not in the currently b b used sub-sample. The second step IV estimator is constructed as an IV estimator in the (1) (2) (3) regression of βi on βi using βi as instrument. That is,

N N b b b (3) (1) 1 (3) (2) − λ = βi βi βi βi . i=1 i=1 X X b b b b b In order to make inferences on λ, one needs to obtain the asymptotic distribution of sample covariances between diﬀerent ﬁrst step estimates. Statements like (1) and (2) are instrumental to accomplish this.

~~The conﬁguration in ΞN,T and a need for statements (1) and (2) occur in other situations as well.~~

Example 4. Pesaran & Yamagata (2018) suggest a new test for factor pricing models that allows many portfolios to be considered simultaneously (with N and T both diverging to inﬁnity). The hypothesis of interest H0 : αi = 0 for all i =1, ..., N, where αi is a pricing error for the portfolio i. To estimate the pricing errors the authors use OLS estimates αi.

A large number of portfolios N does not allow one to establish join Gaussianity of all αi or to consistently estimate their covariance. Pesaran & Yamagata (2018) propose to testb b the hypothesis of interest using statistics based on a weighted sum of squares of αi. They create a properly normalized statistic of the form b N α2 i 1 , σ2 − i=1 i X b 2 where σi are variances of pricing errors. This statistic is directly related to the sample variance of the ﬁrst step estimator, and a statement of its asymptotic Gaussianity directly follows from (1) by the same logic as stated above. Pesaran & Yamagata (2018) develop a CLT for quadratic forms that can be applied in this setting. They make an assumption that the idiosyncratic components can be ﬁltered to make them cross-sectionally independent.3 Here we propose an alternative version of CLT that can be applied under less restrictive assumptions on the cross-sectional dependence of eit’s.

~~3See Assumptions 2 and 3 in Pesaran & Yamagata (2018).~~

7 Example 5. A data-rich IV environment of Bai & Ng (2010) is another example where our linear-quadratic CLTs can be useful. The authors consider an IV setup with many instruments in a panel, where the number of instruments, N, is potentially higher than the number of observations, T . The instruments are generated by a factor model zit =

λi′ Ft +eit, with Ft and eit independent of the structural error εt, and time-series and cross- sectional dependence in eit is allowed, though restricted. Bai & Ng (2010) consider the bias-corrected GMM estimator that corrects for inconsistency of the baseline GMM. It is consistent when N/T = O(1), however, its asymptotic Gaussianity is established under a more restrictive assumption when N/T = o(1). The challenge is that when N/T = O(1), the asymptotic expansion for this estimator has, in addition to a linear term, a quadratic form in the idiosyncratic components eit similar to the second component of ΞN,T . Thus, using statement (1), an asymptotic theory could be developed for the bias-corrected GMM estimator without having to impose N/T = o(1).

In most of these examples, the set of idiosyncratic components e , i = 1, ..., N, t = { it 1, ..., T cannot be regarded independent and/or identically distributed. In most realistic } applications, one is usually willing to assume that eit do not have a strong (detectable) factor structure, but still allow for some correlation between different units, which would not affect consistency. For example, it is reasonable to think that stocks of firms in the same industry or of the same size may react to some local shocks and be correlated, though when averaged over all stocks (and all industries), this co-movement of returns would have no first-order impact on estimation. Our attempt to be agnostic with regard to possible cross-sectional correlation among errors and to avoid explicit modeling of its structure whenever possible comes at a cost of more restrictive time series assumptions. In many applications of interest, it is more credible to impose independence assumptions in a time-series direction rather than in a cross-sectional direction. For example, the efficient market hypothesis implies mean non-predictability of excess returns given past history, which is equivalent to a martingale difference property for the errors. The definition of shocks in macroeconomics similarly presumes their time-series independence. In this paper, we assume time-series independence, which in some cases may be weakened to the martingale difference property or stationarity with some proper mixing condition, but we do not pursue this generalization here.

~~8 3 CENTRAL LIMIT THEOREM~~

In this paper we consider asymptotics as both cross-sectional and time-series sample sizes, N and T , increase to infinity. We allow the data-generating process for all variables to vary with N and T . Define to be a σ-algebra that contains at least the σ-algebras F generated by the full set of variables v ,s = 1, ..., and w ,s,t = 1, ..., for all { s ∞} { st ∞} s and t. It may potentially also contain other events related to common shocks and variables, as long as Assumption 3 stated below is satisfied. We treat γi as non-random k 1 vectors. γ × In order to simplify the notation, in what follows we will denote C to be a positive generic constant, independent of N and T, which may be different in different equations, but does not depend on or change with N or T . We will use the following notation: for a square matrix A, we denote by tr(A) its trace, by maxev(A) – its maximal eigenvalue, and by dg(A) a diagonal matrix of the same size with the elements from the diagonal of A; is the l norm for a vector or the operator norm for a matrix. k·k 2

~~Assumption 1 The random kv-vector vs and kw-vector wst are measurable with respect to the σ-algebra for all s, t, and F~~

~~1 T 1 T (i) E (v v′ ) Ω and 2 E (w w′ ) Ω , where Ω and Ω are T s=1 s s → v T s=1 t~~

~~4 4 (ii) max1 s T E vs~~

~~Assumption 3 (i) Conditional on , the random N-vectors e = (e , ..., e )′ are F t 1t Nt serially independent, and E(e )=0 for all t; t|F E 4 (ii) max1 i N,1 t T (eit)~~

~~Assumption 1 imposes very mild restrictions on the time-series behavior of the com-~~

~~mon (non-entity speciﬁc) variables. For example, the part related to vt is trivially satisﬁed~~

9 if a time series equal to vtvt′ is weakly stationary with summable auto-covariances. As- sumption 2 restricts the influence of any one entity in the cross-sectional average and will eventually contribute to asymptotic negligence of the cross-sectional summands needed for the CLT. Assumption 3(i) is a restrictive assumption which imposes discipline on the time-series structure, and the restriction E(e ) = 0 is a form of strict exogeneity in the t|F first step regression. Uniform moment boundedness in Assumption 3(ii) is traditional. Apparently, Assumptions 1, 2 and 3 are insufficient to establish a central limit theorem, and we need to put some restrictions on the cross-sectional dependence and dependence between idiosyncratic errors and common variables. Indeed, we will use a change of summation ordering:

~~N 1 T v 1 N γ e 1 √T s=1 s √N i=1 i is ξi = , √  1 T 1 N  N i=1 P wst P eiteis X T s=1 t~~

conditional heteroscedasticity in eit that can be related to some common variables from producing dependence in higher-order conditional moments. This ﬂexibility comes at F the cost of imposing some structure on the cross-sectional behavior of eit.

~~3.1 Independence from common variables~~

Assumption 4 (i) The errors et = (e1t, ..., eNt)′, t = 1,...,T are independent from the σ-algebra and identically distributed across t; F E (ii) For the N N covariance matrix N,T = (etet′ ), lim supN,T max ev ( N,T ) < , × E →∞ E ∞ and 1 tr 2 a< ; N EN,T → ∞ 1 (iii) γ′ γ Γ , where Γ is full rank; N EN,T → σ σ 1 N N N N (iv) 2 E (e e e e ) < C. N i1=1 i2=1 i3=1 i4=1 | i1t i2t i3t i4t | P P P P 10 Theorem 3.1 Under Assumptions 1, 2, 3 and 4, the central limit theorem stated in ΣV 0 equation (1) holds with Σξ = , where ΣV =Γσ Ωv and ΣW = aΩw. 0 ΣW ⊗ Numerous papers that establish inferences in factor models commonly assume that the set of factors is independent from the set of idiosyncratic errors, as in Assumption 4(i), though cross-sectional dependence of errors is allowed; see, for example, Assumption D in Bai & Ng (2006). We intended for the first part of Assumption 4(ii) to impose weak cross-sectional dependence as expressed by the covariance matrix; in particular, it means that no strong factor structure is left in the errors; similar assumptions appear in Onatski (2012) and Bai & Ng (2006). The convergence of the trace in Assumptions 4(ii) and 4(iii) is needed for the asymptotic covariance matrix to be properly defined. Assumption 4(iv) is another way to restrict pervasive dependence in multiple variables, in particular, precluding outliers to realize in too many error terms simultaneously. For example, imagine that the cross-sectional dependence is induced by several groups with a factor structure, e.g., stock returns are correlated because there are industry-specific shocks and geography-specific shocks. Imagine that there are a finite number, say G, groups, indexed by g =1, ..., G, which may be overlapping, with each having independent

~~shocks fg,t at time t. Stock i has non-zero loading πi,g only if it belongs to group g. Let the set of groups, to which i belongs, be denoted by G(i). That is,~~

eit = πi,gfg,t + ηit, g G(i) ∈X where ηit’s are independent both cross-sectionally and across time and have ﬁnite fourth cumulants. Then, Assumption 4(iv) is essentially equivalent to the following two conditions: E(f 4 )

11 Another CLT for quadratic forms for time series data can be obtained using Bhansali et al. (2007). The book by Giraitis et al. (2012) has a chapter on this subject and allows for long memory time series as well.

~~3.2 Conditional heteroscedasticity~~

Assumption 4(i) of independence is much stronger than Assumption 3(i) about exogeneity: it does not allow higher conditional moments of eit to co-move with the common variables; in particular, it imposes conditional homoscedasticity. It may be especially problematic in ﬁnancial applications where time-varying volatility is of strong empirical relevance, and returns on many stocks display patterns of changing volatility driven by some common variables. The assumptions below allow for conditional heteroscedasticity.

~~Assumption 5 The errors eit have the following weak (unobserved) factor structure:~~

~~eit = πi′ft + ηit, where the following assumptions hold:~~

(i) The k 1 process f , where k is ﬁxed, is serially independent, conditionally on f × t f E E E 4 4 , with (ft )=0, (ftft′) = Ikf , max1 t,s T ( vs + 1) ft < C, and F |F ≤ ≤ k k k k 4 8 max1 s,t,t∗ T E wst∗ ft

T 2 E 1 (fsfs′) (vsvs′ ) Σfv 0.  T ⊗ −  → s=1 X   Theorem 3.2 Under Assumptions 1, 2, 3 and 5, the statement of the central limit the- Σ 0 orem stated in equation (1) holds with Σ = V , where Σ = ω4Ω and ξ 0 Σ W w W Σ = Γ′ I Σ (Γ I )+Γ Ω . V πγ ⊗ kv fv πγ ⊗ kv ω ⊗ v

12 An interesting feature of this example is that it allows the errors to be weakly cross- sectionally dependent to the extent that they may possess a weak (latent) factor structure. E The condition (ftft′)= Ikf is a normalization and involves no loss of generality. Assump- tion 5(ii) forces the factors to be weak to such an extent that the factor structure cannot be consistently detected; it implies that the covariance matrix of idiosyncratic errors would satisfy the ﬁrst half of Assumption 4(ii). Moreover, this factor structure may be closely related to the common variables in , which causes the cross-sectional dependence F among the errors eit to change with the common variables and allows a very ﬂexible form of conditional heteroscedasticity. Indeed, the conditional cross-sectional covariance is

~~E E I 2 (eitejt )= πi′ (ftft′ )πj + i=j ωi . |F |F { }~~

Since we do not restrict E(f f ′ ) beyond proper moment conditions, the strength of any t t |F cross-sectional dependence as well as error variances may change stochastically depending on realizations of the common variables. The moment conditions in Assumption 5(i) help to establish asymptotic negligibility

~~of the time-series summands. Assumption 5(iii) about Γω and Assumption 5(iv) allow us to deﬁne properly the asymptotic covariance matrix.~~

~~4 VALID INFERENCE~~

In this Section, we ﬁrst discuss estimation of asymptotic variances for asymptotic inference when this leads to valid inference. Then, we propose alternative tools based on the wild bootstrap to apply in situations when asymptotic inference fails to provide asymptotically correct inference.

~~4.1 Asymptotic inference~~

Statistical inferences such as conﬁdence set construction and hypotheses testing about the structural parameter typically require consistent estimation of asymptotic variances of all important quantities that are asymptotically Gaussian. The easiest to implement and thus the most appealing from an applied perspective are those that use the same variables and have a structure similar to the original averages, such as the statement in equation (2). Notice that equation (2) contains the cross-sectional summation outside, and hence it treats the cross-section as nearly uncorrelated observations, or at least it ignores the cross-sectional correlation. A relevant analogue is the diﬀerence between the long-run

13 covariance and instantaneous covariance in a classical time series. However, implementing an analogue of long-run covariance estimation here would be a challenge since we do not have any cross-sectional stationarity or a measure of distance between cross-sectional entities. Rather, we explore under which conditions the convergence in (2) holds. Theorem 4.1 below obtains a statement for the case when the common variables are independent from the idiosyncratic errors, while Theorem 4.2 establishes a similar statement for the conditionally heteroscedastic case.

~~Theorem 4.1 If in addition to Assumptions 1, 2, 3, 4 we also have that~~

~~dg( ) 0 as N, T , (4) kEN,T − EN,T k→ →∞ then consistency statement (2) holds.~~

~~Theorem 4.2 If in addition to Assumptions 1, 2, 3, 5 we also have that Γπγ = 0, then consistency statement (2) holds.~~

The additional assumption (4) in Theorem 4.1 strengthens conditions on the weakness of the cross-sectional correlation; in particular, it requires that the covariance matrix converges to a diagonal one. The additional assumption in Theorem 4.2 requires that the weights used for averaging the cross-sectional entities are orthogonal to the loadings on the latent factor structure, which precludes the latent factor structure (that represents the cross-sectional dependence) from being ampliﬁed. This is a necessary assumption for consistency of the variance estimator. Indeed, let assumptions 1, 2, 3, 5 hold, and consider the ﬁrst component of ξi:

(1) 1 ξ = v γ e = π γ′Υ +˜η , i √ t i it i i T i T t X whereη ˜ = 1 v γ η , and Υ = 1 T f v . Note that 1 N η˜ (0, σ2) and i √T t t i it T √T t=1 t t √N i=1 i η p ⇒N 1 N η˜2 σ2, as all conditions of Theorems 3.2 and 4.2 are satisﬁed by cross-sectionally N i=1 i → Pη P P andP time independent errors ηis. Assumption 5(iv) guarantees that ΥT (0, Σfv) as 1 N ⇒ N T , while according to Assumption 5(ii), we have π γ′ Γ as N . → ∞ √N i=1 i i → πγ → ∞ Thus, N P 1 (1) 2 ξ 0, Γ Σ Γ′ + σ , √ i ⇒N πγ fv πγ η N i=1 X 1 N (1) 2 p 2 1 N while ξ σ because π γ′γ π′ 0 by Assumptions 2 and 5(ii). N i=1 i → η N i=1 i i i i → P P

~~14 4.2 Bootstrap inference~~

1 N As one way to conduct valid inferences in settings when N i=1 ξiξi′ is an inconsistent estimator of the variance (in particular, when Γ = 0 under Assumption 5), we propose πγ 6 P the following simple wild bootstrap procedure. In each bootstrap repetition,

~~(i) simulate independent draws of random variables δ +1, 1 , t = 1, ..., T, with t ∼ { − } success probability 1/2;~~

~~(ii) compute the bootstrap analogues of errors eit∗ = δteit, i =1, ..., N, t =1, ..., T ;~~

~~(iii) compute the bootstrap analogues ξi∗ using the deﬁnition of ξi, with eit∗ in place of~~

~~eit for all i =1, ..., N, t =1, ..., T.~~

Then the distribution of Ξ = 1 N ξ has the same asymptotic limit as that of Ξ N,T∗ √N i=1 i∗ N,T has, and can be used for inferences.P Alternatively, the distribution ofΞN,T∗ normalized by 1 N the bootstrap analogue of the variance estimate N i=1 ξi∗ξi∗′ can be used to approximate the distribution of Ξ normalized by the variance estimate 1 N ξ ξ . We call the N,T P N i=1 i i′ two described bootstrap procedures bootstrap and bootstrap-t. InP the next Section, we implement both variations of the wild bootstrap for the setup of Assumption 5. The wild bootstrap works because it introduces independence in the time direction while preserving the (unknown) cross-sectional dependence. Speciﬁcally, we base our proof of Theorem 3.2 on the change of order of summations in double/triple summations over i and time index/indices. For example,

1 N 1 T 1 N Ξ(1) = ξ(1) = v γ e . N,T √ i √ √ t i it N i=1 T t=1 N i=1 ! X X X We then argue that our assumptions guarantee that the CLT with respect to summation (1) over t is applicable. If the assumptions of the CLT hold, then ΞN,T is asymptotically Gaussian with mean zero and variance equal to the limit of 1 T E 1 v γ e 2 . T t=1 √N i t i it This limit clearly depends on how much there is cross-sectionalP correlationP betweeneit and ejt. Now, in the bootstrapped samples,

N T N (1) 1 (1) 1 1 Ξ ∗ = ξ ∗ = v γ e δ . N,T √ i √ √ t i it t N i=1 T t=1 N i=1 ! X X X Conditional on the original sample, only δt’s are random, and they are independent and have zero mean and unit variance. When T is large, this bootstrapped normalized sum

15 satisﬁes the CLT and thus converges to a zero mean Gaussian random variable with 2 the variance equal to the limit of 1 T 1 N v γ e as N, T . This limit T t=1 √N i=1 t i it 2 → ∞ coincides with 1 T E 1 N v γ e under relatively weak assumptions, as long T t=1 √N i=1 tPi it P as the Law of LargeP Numbers P holds. For example, under Assumptions 5, we have: 2 2 1 T 1 N 1 T 1 N lim v γ e = lim Γ v f + v γ η T √ t i it T πγ t t √ t i it t=1 N i=1 ! t=1 N i=1 ! X X p X 2 X Γ Σ Γ′ + σ . → πγ fv πγ η A similar argument can be made about the second component as well. Indeed,

N T N (2) 1 (2) 1 1 Ξ ∗ = ξ ∗ = w e e δ δ . N,T √ i T √ st it is t s N i=1 s=1 texperiments in the following Section, we check, among other things, that both wild bootstrap variations deliver correctly sized tests even when the asymptotic- t tests fail to do so, and also that the bootstrap-based tests have a non-trivial power.

~~5 MONTE CARLO SIMULATIONS~~

The goals of this Section are to check ﬁnite sample performance of an asymptotic Gaussian 1 N approximation for ΞN,T , to explore when the variance estimator Σξ = N i=1 ξiξi′ allows to construct reliable asymptotic-t inferences, and to evaluate the performanceP of the wild b bootstrap and bootstrap-t procedures in terms of both size and power.

~~5.1 Setup~~

~~Our setup adheres to Assumption 5. We generate the errors eit according to the following weak (unobserved) factor structure:~~

eit = πi′ft + ηit, where f iid (0, 1) across t =1, ..., T, γ = 1 for all i =1, ..., N, and η = ω ǫ , where t ∼ N i it i η,it ǫ are iid (0, 1) across i and t. The standard deviations are set to ω = c (1 + τ ) , η,it N i ω | i|

16 and the factor loadings are π =(c + τ ) /√N, where τ iid (0, 1) across i =1, ..., N. i π i i ∼ N The multiplier cω is tuned so that the average cross-sectional variance of ηit is unity. The parameter cπ indexes the degree of cross-sectional dependence as measured by the strength N 2 of the factor structure. Speciﬁcally, as π π′ 1+ c , this parameter is assumed to i=1 i i → π be bounded for the Gaussian approximationP to hold. We also have Γπγ = cπ, hence we can expect consistency of variance estimation only when cπ = 0, so we will explore the distortions for diﬀerent values of cπ. The errors generated this way are cross-sectionally dependent and heteroscedastic, while still satisfying Assumption 5. The common variables are generated as follows: v = c f + 1 c2 ǫ , with ǫ t fv t − fv v,t v,t ∼ iid (0, 1) across t =1, ..., T, and wst = vsvt,s,t =1, ..., T. All theq disturbances ft, τi, ǫη,it N and ǫv,t are mutually independent. The parameter cfv indexes the dependence between

~~common variables and eit. The mean zero Assumption 3(i) requires cfv = 0; the non-zero~~

values of cfv index deviations from the null hypothesis E(ξi)=0, and will be used to study the power properties of the proposed wild bootstrap. In wild bootstrap samples, 1 we generate the bootstrap errors by e∗ = δ e , where δ = 2ζ 1, and ζ iid ( ) it t it t t − t ∼ B 2 across t, ..., T . The bootstrap analogues of γi, vt and wst are set equal to their original sample values. The distribution characteristics are computed from 10,000 simulations, while the rejection rates are based on 5,000 simulation runs. In all simulations, we set N = T = 500. The number of bootstrap repetitions is 600.

~~5.2 Results~~

Table 1 contains distributional characteristics of ΞN,T . We report averages, coeﬃcients of skewness, coeﬃcients of kurtosis, and right 5% quantiles of normalized marginal distributions of both elements of ΞN,T . For the exactly normal distributions, these values are 0, 0, 3 and 1.645, respectively.

The actual distribution of the linear component of ΞN,T is very close to Gaussian, in all respects: all the moments and the right tail are almost equal to their theoretical counterparts. The quadratic component of ΞN,T , however, albeit mean unbiased, is some- what positively skewed and a bit leptokurtic. The shifted right quantile conﬁrms slight over-dispersion. The distortions, however, do not seem to increase with the strength of the error factor structure. In Table 2 we document the empirical rejection rates for tests with 10%, 5% and 1% declared size based on the asymptotic-t, wild bootstrap and wild bootstrap-t approaches.

~~17 Table 1: Characteristics of simulated ﬁnite-sample distribution of ΞN,T for N = T = 500~~

element of Ξ linear quadratic N,T → c mean skew kurt quant mean skew kurt quant π ↓ 0.5 0.01 0.05 3.01 1.63 0.01 0.25 3.20 1.71 1 0.01 −0.04 2.97 1.64 −0.01 0.25 3.20 1.71 2 0.00 −0.00 2.95 1.65− 0.02 0.20 3.11 1.71 − Notes: Based on 10,000 simulations. The first component of ΞN,T is labeled ‘linear’ and second component is labeled ‘quadratic’. ‘Mean’ stands for average, ‘skew’ for skewness coefficient, ‘kurt’ for kurtosis coefficient, and ‘quant’ for right 5% quantile of simulated marginal distribution of components of ΞN,T normalized by corresponding standard deviations. Rows with cπ = 0.5, 1 and 2 correspond to a very weak, weak and moderately strong error factor structure.

In the asymptotic-t approach we create a t-statistic using Σξ as a variance estimator and compare it with the symmetric standard Gaussian critical values. We also explore the b performance of two wild bootstrap procedures – one that bootstraps ΞN,T and another that bootstraps the t-statistics (referred to in Table 2 as bootstrap ξ and bootstrap t). In both bootstrap procedures, we compare the absolute value of the statistic from the sample to the right quantile of the absolute value of the bootstrapped statistic. We verify the empirical size by setting cfv = 0 and empirical power by setting cfv = 0.1 for relatively small deviations from the null and cfv =0.2 for relatively large deviations from the null. As expected, the size of the test based on asymptotic approximations sometimes deviates from nominal rates by a wide margin, especially for the linear component of

ξN,T , the gap quickly increasing with the strength of the error factor structure. This happens due to inconsistency of the variance estimator Σξ and becomes more pronounced with stronger cross-sectional dependence. What is surprising is that for the quadratic b component of ξN,T , the distortions are relatively minor and not very sensitive to the strength of the factor structure. In contrast, both bootstrap procedures exhibit excellent size control and stability thereof across the strength of the error factor structure for both components of ξN,T . In terms of power, however, the two bootstrap statistics are approximately equally powerful for the linear component of ξN,T , while there is a gap, sometimes sizable, between power ﬁgures for its quadratic component. It seems that bootstrapping the statistic itself is preferable.

18 Table 2: Simulated rejection rates for asymptotic and wild bootstrap tests element linear quadratic → 1% 5% 10% 1% 5% 10% c asy bootstrap asy bootstrap asy bootstrap asy bootstrap asy bootstrap asy bootstrap π ↓ t ξ t t ξ t t ξ t t ξ t t ξ t t ξ t

Size: cfv =0 0.5 21 1 85 5 141010 01 1 36 5 61010 1 71 1 165 5 241010 01 1 36 5 61010 2 251 1 385 5 461010 0 1 1 3 6 6 71112 19 Power: cfv =0.1 0.5 9 6 6 2217 17 3227 26 0 2 1 3 7 6 712 12 1 3916 16 5635 35 6747 46 1 2 2 3 7 6 6 13 11 2 7929 29 8751 51 9063 63 1 6 3 5 14 9 1120 17

~~Power: cfv =0.2 0.5 35 28 26 58 51 49 68 63 62 1 5 3 5 13 9 1019 16 1 90 73 71 96 88 87 97 93 93 2 10 5 9 21 15 17 30 25 2 100 93 92 100 98 98 100 99 99 18 45 29 40 61 50 53 68 61~~

~~Notes: The table contains actual rates, computed from 5,000 simulations for 10%, 5% and 1% declared size asymptotic-t (‘asy t’), bootstrap~~

~~(‘bootstrap ξ’) and bootstrap-t (‘bootstrap t’) two-sided tests for deviations of each component of ΞN,T from the zero value. The ﬁrst component~~

~~of ΞN,T is labeled ‘linear’ and second component is labeled ‘quadratic’. The size ﬁgures are in panel with cfv = 0, and power ﬁgures are in~~

~~panels with cfv =0. Rows with cπ =0.5, 1 and 2 correspond to a very weak, weak and moderately strong error factor structure. 6 6 CONCLUDING REMARKS~~

Possible directions for future research may be relaxing the error time-series independence to martingale diﬀerence structures and inventing ways to consistently estimate the asymptotic variance matrix when it is not diagonal in the limit. Other interesting areas involve establishing formal properties of the proposed wild bootstrap schemes, exploring the pos- sibility of asymptotic reﬁnements, and examining the superiority of one bootstrap scheme over the other.

~~A APPENDIX: PROOFS~~

~~A.1 Preliminary results~~

~~We use the following central limit theorem for a vector valued martingale diﬀerence sequence:~~

Lemma A.1 Let the sequence (Z , ), t =1, .., T, be a martingale diﬀerence sequence t,T Ft,T of r 1 random vectors with Σ = var T Z . If the following two conditions hold × T t=1 t,T as T , →∞ P

2 T 4 (1) (min ev(Σ ))− E Z 0, T t=1 k t,T k → 2 P T 2 (2) (min ev(Σ ))− E Z Z′ Σ 0, T t=1 t,T t,T − T → P then, as T , →∞ T 1/2 Σ− Z (0,I ). T t,T ⇒N r t=1 X Proof of Lemma A.1 Indeed, the statement of Lemma A.1 holds if for any non- 1/2 T random r 1 vector λ, we have (λ′Σ λ)− λ′Z (0, 1). Let us define a scalar × T t=1 t,T ⇒N martingale difference sequence z = λ Z with variance σ2 = var T λ Z = λ Σ λ. t ′ t,T P T t=1 ′ t,T ′ T Let us check that all conditions of the central limit theorem by HeydP e & Brown (1970) are satisfied for δ = 1. Indeed,

~~T T T 1 4 1 4 1 4 4 E z = E λ′Z λ E Z 0, σ4 | t| (λ Σ λ)2 | t,T | ≤ ( λ 2 min ev(Σ ))2 k k k t,T k → T t=1 ′ T t=1 T t=1 X X k k X~~

~~20 and 2 2 T 2 T 2 E t=1 zt E t=1(λ′Zt,T ) 2 1 = 1  σT −   λ′ΣT λ −  P P~~

   T  2 1 E ′ ′ = 2 λ Zt,T Zt,T ΣT λ (λ′ΣT λ)  − !  t=1 X  T  2 1 4 E ′ 2 2 λ Zt,T Zt,T ΣT 0. ≤ ( λ min ev(ΣT )) k k  −  → k k t=1 X 1 T   These two conditions imply that σ− zt (0, 1). This ﬁnishes the proof. T t=1 ⇒N As a preliminary result, we establishP a central limit theorem for quadratic forms. The idea of this result comes from the CLT for quadratic forms by de Jong (1987). All random variables are implicitly indexed by the sample sizes T (or N, T in the further application to factor models), which are omitted to reduce clutter; for example, Wst in full notation

~~is indexed as Wst,T or Wst,N,T .~~

Lemma A.2 Let Wst = Wst(Xst, es, et) be a set of random vectors deﬁned for all s > t, where s, t 1, ..., T , such that X is a random vector measurable with respect to the ∈ { } st σ-algebra , and all e are independent from each other, conditionally on . Assume that F t F E(W , e )=0 and E(W , e )=0. (A.1) st|F t st|F s T Deﬁne W (T )= s=1 t

Lemma A.3 Let Wst = Wst(Xst, es, et) satisfy all conditions of Lemma A.2. Let Vs = V (X , e ) be a random vector deﬁned for all s 1, ..., T such that X is a random vector s s s ∈{ } s measurable with respect to the σ-algebra , and E(V )=0. Deﬁne V (T ) = T V F s|F s=1 s and Σ = var(V (T )). Assume the following statements hold as T : V,T →∞ P

~~21 (a) Σ Σ , where Σ is a full rank matrix; V,T → V V 4 (b) T max1 s T E Vs 0; ≤ ≤ k k → 2 T (c) E V V ′ Σ 0; s=1 s s − V,T → P 3 E (d) T max 1 t~~

~~stated in de Jong (1987). Call Wst clean if~~

E (W W .... W )=0 s1t1 ⊗ s2t2 ⊗ sktk when at least one index from the set s , t , ..., s , t has a value that occurs only once. { 1 1 k k} The functional form of Wst and the condition stated in (A.1) guarantee that in our case

~~Wst is clean. Indeed, if, for example, the index s1 occurs only once, then~~

E W W .... W = E E(W W .... W , e , e , e , ..., e ) s1t1 ⊗ s2t2 ⊗ ⊗ sktk s1t1 ⊗ s2t2 ⊗ ⊗ sktk |F t1 s2 t2 tk E E = (Ws1t1 , et1 , es2 , et2 , ..., etk ) Ws2t2 .... Wsktk |F ⊗ ⊗ ⊗ E E = (Ws1t1 , et1 ) Ws2t2 .... Wsktk =0. |F ⊗ ⊗ ⊗ Now, W (T )= T W = T Z , where Z = W . We denote by the s=1 t ~~0. Now let us check condition (1) of W,T → Lemma A.1:~~

~~4 4 E Zs,T = E Wst k k   t~~

22 The last statement follows from the fact that Wst is clean, and non-zero summands are only those where either t = t = t = t or the set t , t , t , t consists of 1 2 3 4 { 1 2 3 4} two distinct elements each occurring twice. We also notice that E W 2 W 2 k st1 k k st2 k ≤ 1 E 4 E 4 E 4 4 Wst1 + Wst2 max1 t,s T Wst < CT − due to condition (ii). 2 k k k k ≤ ≤ ≤ k k 4 2 T 4 1 Hence, E Z CT − . Thus, E Z CT − , implying that condition k s,T k ≤ s=1 k s,T k ≤ (1) of Lemma A.1 holds. P Now let us turn to condition (2). First, notice that

~~T T~~

~~ΣW,T = var(W (T )) = var Wst = var(Wst), s=1 t~~

~~T 2 E Z Z′ Σ  s,T s,T − W,T  s=1 X F   2 T ′ E = Wst1 Wst2 ΣW,T  ! ! −  s=1 t1~~

~~T 2 T T E E Wst1 Wst′ 2 = tr Ws1t1 Ws′1t2 Ws2t3 Ws′2t4   s=1 t1=t2~~

~~23 3 (iv), and a category when s1 = s2, there being at most CT of such summands, each 4 4 smaller than C max1 t,s T E Wst < CT − . Thus, ≤ ≤ k k~~

T 2 E Wst1 Wst′ 2 0. (A.3)   → s=1 t1=t2 F X X6   Putting statements (A.2) and (A.3) together we obtain that condition (2) of Lemma A.1 is satisﬁed. Thus, the conclusion of Lemma A.2 holds.

~~Proof of Lemma A.3. Let us deﬁne Z =(V ′, W ′ )′, and let be deﬁned as in s s t~~

E V W ′ = E E(V W ′ , e ) = E V E(W ′ , e ) =0. s st s st|F s s st|F s Thus, T ΣV,T 0 ΣV 0 ΣT = var Zs = . 0 ΣW,T → 0 ΣW s=1 ! X The right-hand-side is a full rank matrix by condition (i) of Lemma A.2 and condition

~~(a) of Lemma A.3. Thus, the minimal eigenvalue of ΣT is separated away from zero for large T . Now,~~

~~T T T 4 E Z 4 C E V 4 + C E W . k sk ≤ k sk  st  s=1 s=1 s=1 t~~

~~T 2 T 2 E E ZsZs′ ΣT ZsZs′ ΣT  −  ≤  −  s=1 s=1 F X X    T 2 T 2~~

~~= E V V ′ Σ +2E W V ′  s s − V,T   st s  s=1 s=1 t~~

~~24 Here we use that the Frobenius norm of a matrix equals to the sum of squares of all elements and can be decomposed into sums over four blocks of the matrix. Condition (c) guarantees that~~

~~T 2 T 2 E E VsVs′ ΣV,T C VsVs′ ΣV,T 0.  −  ≤  −  → s=1 F s=1 X X     During the proof of Lemma A.2 we show that~~

~~2 T ′ E Wst Wst ΣW,T 0.  ! ! −  → s=1 t~~

~~T 2 T T E E Wst Vs′ = tr Ws1t1 Vs′1 Vs2 Ws′2t2  !  s=1 t~~

~~Lemma A.4 For an N N symmetric matrix A = (a ) denote to be the Hadamard × ij ⊙ product. Then A A √N A 2. k ⊙ k ≤ k k~~

~~Proof. Using the equivalence of norms, we have~~

~~4 2 2 √ 2 A A A A F = aij max aij aij A A F N A . k ⊙ k≤k ⊙ k ≤ 1 i,j N ≤k kk k ≤ k k 1 i,j N ≤ ≤ 1 i,j N s ≤X≤ r s ≤X≤~~

~~A.2 Proofs for Independent case~~

~~Proof of Theorem 3.1. We will check that all conditions of Lemma A.3 are satisﬁed for 1 N 1 e e W = w e e = w t′ s st √ st it is T st √ T N i=1 N X~~

~~25 and 1 N 1 γ e V = γ e v = ′ s v . s √ i is ⊗ s √ √ ⊗ s T N i=1 T N X (i) First notice that due to Assumption 4(ii)~~

~~e e 2 tr 2 E t′ s = EN,T a. √ N → " N #~~

~~Due to the independence between the common variables and eit and because Wst is clean, we have:~~

~~T T 2 1 et′ es Σ = var W = E(w w′ )E aΩ , W,T st T 2 st st √ → w s=1 t~~

~~(ii) By Assumption 4(i) and the i.i.d. nature of et, we have:~~

~~4 N N N N 4E 4 E 4 E et′ es C E 2 T Wst = wst (ei1tei2tei3tei4t) < C. k k k k √N ≤ N 2 " # i1=1 i2=1 i3=1 i4=1 X X X X E E 4 Here we used that (ei1tei2tei3tei4t) max1 i N,1 t T (eit)~~

~~T T 2 1 es′ et 1 2 W W ′ Σ = w w′ tr( ) st st − W,T T 2 st st √ − N EN,T s=1 t~~

hence it is enough to prove that E A 2 0 and E A 2 0. The latter is postulated k 1k → k 2k → by Assumption 1(iii). Notice that all summands inA1 are uncorrelated with each other due to Assumptions 3(i) and 4(i). Thus,

~~2 T 2 2 1 4 es′ et tr( N,T ) E tr(A1A′ ) = E wst E E 1 T 4 k k  √N − N  s=1 t~~

~~26 (iv) If the set s ,s , t , t contains four distinct indexes, then { 1 2 1 2}~~

4 4 tr E es1 es′ 1 et1 et′1 es2 es′ 2 et2 et′2 E ′ ′ E T Ws1t2 Ws2t1 Ws2t2 Ws1t1 wst 2 ≤ k k N C 4 C 4 C tr N max ev 0. ≤ N 2 EN,T ≤ N 2 EN,T ≤ N → We now move to conditions (a)-(d) of Lemma A.3. (a) By Assumptions 4(iii) and 1(i) we have

1 1 T Σ = γ′ γ E (v v′ ) Γ Ω , V,T N EN,T ⊗ T s s → σ ⊗ v s=1 ! X and the limit is a full rank matrix. (b) Next, 4 4 1 1 4 T E V = E γ′e E v , k sk T √ s k sk " N # where E v 4 C due to Assumption 1(ii). Assumptions 2 and 4(iv) imply that k sk ≤ 4 N N N N E 1 1 E γ′es = (ei1tei2tei3tei4t) γ′ γi2 γ′ γi4 √N N 2 i1 i3 " # i1=1 i2=1 i3=1 i4=1 X X X X N N N N 4 1 E < max γi (ei1tei2tei3tei4t) < C. 1 i N k k N 2 | | ≤ ≤ i1=1 i2=1 i3=1 i4=1 X X X X (c) Next,

~~T 1 1 T V V ′ Σ = γ′ γ v v′ E (v v′ ) s s − V,T N EN,T ⊗ T s s − s s s=1 s=1 ! X T X 1 γ′eses′ γ γ′ N,T γ + E (v v′ ) T N − N ⊗ s s s=1 X = A1 + A2.~~

~~Notice that A1 and A2 are uncorrelated, hence~~

~~T 2 E E E VsVs′ ΣV,T = tr( (A1′ A1))+tr( (A2′ A2)) .  −  s=1 F X  ~~

~~27 Assumption 1(iv) guarantees the convergence of the ﬁrst term. Notice that the summands~~

~~in A2 are uncorrelated due to time independence of errors, hence~~

~~T 2 1 γ′eses′ γ 1 4 tr(E(A′ A )) = E γ′ γ E v 2 2 T 2 N − N EN,T k sk s=1 " # X 2 C γ′eses′ γ 1 E γ′ γ . ≤ T N − N EN,T " #~~

~~1/2 Given the bounds on the fourth moment of N − γ′es derived in the proof of part (b) we get that condition (c) holds. (d) By Assumption 4(i) we have that~~

~~3 es′ 1 γ γ′es2 es′ 1 et es′ 2 et T E W V ′ V W ′ = E w v′ v w′ E . s1t s1 s2 s2t s1t s1 s2 s2t √ √ √ √ N N N N ~~

~~Using that scalars can be reshuﬄed to make two same-index et stand back to back and employing time series independence of errors, we obtain that~~

es′ 1 γ γ′es2 es′ 1 et es′ 2 et 1 E = tr γγ′E(e e′ )E(e e′ )E(e e′ ) √ √ √ √ N 2 s2 s2 t t s1 s1 N N N N 1 3 C tr( γγ′) max ev . ≤ N 2 EN,T ≤ N 1 Here we use Assumption 2 to get N − tr(γγ′)

~~3 E C T max Ws1tVs′1 Vs2 Ws′2t 0. 1 t~~

~~Proof of Theorem 4.1. We will prove the following three statements for~~

~~1 T ξ = γ e v V,i √ i is ⊗ s T s=1 X and 1 T ξ = w e e : W,i T st it is s=1 t~~

P 2 Let us start with statement (i). Denote by σi the diagonal and by σij the oﬀ-diagonal elements of matrix . Notice that the additional assumption of Theorem 4.1 implies EN,T that N γ′ N,T γ 1 2 Γ = lim E = lim γ γ′σ . σ N N i i i i=1 X 1 N 2 1 T Let us deﬁne Σ = N − γ γ′σ T − E (v v′ ) , and notice that Σ Σ . V,T i=1 i i i s=1 s s V,T → V Thus, e P P e N N T T 1 1 2 ξ ξ′ Σ = (γ γ′e e ) (v v′) I s = t σ (γ γ′) E(v v′) N V,i V,i − V,T NT i i is it ⊗ s t − { } i i i ⊗ t t i=1 i=1 t=1 s=1 X X X X e N T 1 2 2 = (e σ )(γ γ′) (v v′) NT it − i i i ⊗ t t i=1 t=1 X X 1 N T + (γ γ′e e ) (v v′) NT i i is it ⊗ s t i=1 t=1 s=t X X X6 N T 1 2 + γ γ′σ v v′ E (v v′) NT i i i ⊗ t t − t t i=1 t=1 X X = A1 + A2 + A3.

Notice that the three terms are uncorrelated, so it is enough to prove that tr E(A A′ ) j j → 0 for j = 1, 2, 3. Indeed, if the expectation of the Frobenius norm of a matrix converges to zero, this implies that each entry converges to zero as well. First,

N T T 1 2 2 2 2 tr(E(A A′ )) = tr E γ γ′γ γ′ (e σ )(e σ ) (v v′v v′ ) 1 1 N 2T 2 i i j j it − i js − i ⊗ t t s s " i,j=1 t=1 s=1 #! X X X N T 1 2 2 = tr γ γ′γ γ′ cov(e , e ) tr E(v v′v v′) N 2T 2 i i j j it jt t t t t i,j=1 t=1 X X T 1 4 E 2 2 2 E 4 C max γi max (eit σi ) vt . ≤ T 2 1 i N k k 1 i N − k k ≤ T t=1 ≤ ≤ ≤ ≤ X Here we used that eit’s are independent from each other for diﬀerent t by Assumption 3(i), which forces s = t. The last inequality uses Assumptions 1(ii), 3(ii) and 2.

~~29 Consider the term A2 and notice that any two summands in the two-directional sum~~

~~(over t and over s) are uncorrelated due to time series independence of et’s and all summands are mean zero. Thus,~~

1 T N tr(E(A A′ )) = tr E(γ γ′γ γ′ e e e e ) E(v v′v v′ ) 2 2 N 2T 2 i i j j it is jt js ⊗ s t t s t=1 s=t i,j=1 X X6 X N T 1 2 1 = tr(γ γ′γ γ′ σ ) tr E (v v′v v′ ) . N 2 i i j j ij T 2 s t t s i,j=1 t=1 s=t X X X6 2 T E E 4 We notice that T − t=1 s=t tr (vsvt′vtvs′ ) vt

~~30 Let us turn to statement (ii):~~

~~N N T 1 1 2 2 4 ξ ξ′ Σ = e e σ w w′ N W,i W,i − W,T T 2N it is − i st st i=1 i=1 s=1 t~~

Again, A1, A2 and A3 are uncorrelated with each other. Thus, we can deal with each one of them separately. We show that the expectation of the Frobenius norm of each matrix converges to zero, this implies that each entry converges to zero as well. Let us start with

~~1 N T tr(E(A A′ )) = tr E(w w′ w ∗ ∗ w′ ∗ ∗ ) E(b b ∗ ∗ ), 1 1 T 4N 2 st st s t s t i,t,s j,t ,s i,j=1 s,s∗=1 t~~

~~b = e2 e2 σ4 =(e2 σ2)(e2 σ2)+ σ2(e2 σ2)+ σ2(e2 σ2). i,t,s it is − i it − i is − i i is − i i it − i~~

Notice that E(b b ∗ ∗ ) = 0 only if at least one of the indexes from the set t, t∗,s,s∗ i,t,s j,t ,s 6 { } appears twice. Thus, the summation over time index is three-dimensional and there are 3 2 E at most CT N non-zero summands in tr ( (A1A1′ )). Let us bound every summand from above. Notice that since t

First, consider the case when the set s ,s∗, t , t∗,s ,s∗, t , t∗ contains at most three { 1 1 1 1 2 2 2 2} distinct indexes (this makes the summation over time three-dimensional). We can show 4 2 E 4 E 4 2 that each summand is bounded by T − N − max1 t,s T wts max1 i N,1 t T eit ≤ ≤ k k ≤ ≤ ≤ ≤ C/ (T 4N 2) in absolute value, and as there are at most N 2T 3 of them (two-dimensional ≤ cross-sectional and three-dimensional over time summations), the sum of such terms will go to zero.

Finally, we consider the case when the set s ,s∗, t , t∗,s ,s∗, t , t∗ contains four { 1 1 1 1 2 2 2 2} distinct indexes. Then each summand of this type is bounded in absolute value by a 2 b 2 c σij (σi ) (σj ) 4 C | | max E wst , 4 2 1 s,t T T N ≤ ≤ k k where a + b + c = 4, and the values of a, b and c depend on which indices coincide with which; however, due to the conditions s , t = s , t and t < s , t < s , we know { 1 1} 6 { 2 2} 1 1 2 2 that the set s ,s , t , t contains at least three distinct indexes. Thus, c and b are either { 1 2 1 2} 0 or 1 each, and a 2. Hence, due to Assumption 1(ii), the corresponding sum is bounded ≥ above by N N N N C a 2 b 2 c C 4 2 σij (σi ) (σj ) max σi σij N 2 | | ≤ N 2 1 i N i=1 j=1 ≤ ≤ i=1 j=1 X X X X C N C N N = σ4 + σ2 (A.4) N 2 i N 2 ij i=1 i=1 i=j X X X6 C 4 C 2 C max σi + N,T dg( N,T ) F . 1 i N 2 ≤ N ≤ ≤ N kE − E k ≤ N In the first inequality, we use σ σ σ . In the second inequality, we use the definition | ij| ≤ i j of the Frobenius norm. In the last inequality, we use that for any symmetric matrix A, 2 2 we have A N A and assumption stated in Theorem 4.1. Thus, tr (E(A A′ )) 0. k kF ≤ k k 2 2 → Next, Assumption 1(iii) implies the convergence of A3. This finishes the proof of (ii). Finally, we need to prove statement (iii) that

~~N T T 1 p ∗ ′ ∗ 3/2 (γi vs )wsteis eiteis 0. NT ∗ ⊗ → i=1 s=1 t~~

~~32 the six-dimensional summation over time indexes has mostly zeros and can be reduced~~

~~to three-dimensional summation over time indexes as the set s , t ,s∗,s , t ,s∗ should { 1 1 1 2 2 2} have any distinct index to appear at least twice.~~

~~First, consider only those terms for which the set s , t ,s∗,s , t ,s∗ contains at { 1 1 1 2 2 2} 2 2 most two distinct indexes; there are at most N T of such terms. Since t1 < s1 and~~

t2 < s2, each time index can appear at most four times; thus, the highest power of each individual shock can be the fourth. As a result, each summand is bounded above by 2 3 2 E 2 E 2 E 4 3/2 N − T − max1 i N γi max1 t T vt max1 t,s T wst max1 i N,1 t T eit . ≤ ≤ k k ≤ ≤ k k ≤ ≤ k k ≤ ≤ ≤ ≤ Given Assumptions 1(ii) and 3(ii), the sum of these terms is bounded above by C/T . Finally, consider only those terms for which the set s , t ,s∗,s , t ,s∗ contains ex- { 1 1 1 2 2 2} actly three distinct indexes. The summation over these indexes is equal to

N 1 2 2 3 tr γ γ′ C σ σ σ + C σ . N 2 i j 1 ij i j 2 ij i,j=1 ! X 3 2 2 The term σ appears when s , t ,s∗ = s , t ,s∗ , while σ σ σ arises when the sets ij { 1 1 1} { 2 2 2} ij i j s , t ,s∗ and s , t ,s∗ have two coinciding indexes each. Therefore, { 1 1 1} { 2 2 2} N N 1 2 2 1 6 1 2 2 tr γ γ′ σ σ σ = tr γ γ′σ + tr (γ σ )(γ′ σ )σ N 2 i j ij i j N 2 i i i N 2 i i j j ij i,j=1 ! i=1 ! i=j ! X X X6 N 1 2 6 1 2 2 = γ σ + tr(γ γ′ )σ σ σ N 2 k ik i N 2 i j i j ij i=1 i=j X X6 2 1 6 N,T dg( N,T ) 4 max γi max σi + kE − E k max σi 0. ≤ 1 i N k k N 1 i N N 1 i N → ≤ ≤ ≤ ≤ ≤ ≤ Also,

N N 1 3 1 2 6 1 3 tr γ γ′ σ = γ σ + tr(γ γ′ )σ N 2 i j ij N 2 k ik i N 2 i j ij i,j=1 ! i=1 i=j X X X6 N 2 1 6 1 2 2 max γi max σi + σij max σi 0. ≤ 1 i N k k N 1 i N N 2 1 i N → ≤ ≤ ≤ ≤ i,j=1 ≤ ≤ ! X 2 N 2 Here we used the statement N − σ 0, which is proved in equation (A.4). This i,j=1 ij → ends the proof of Theorem 4.1. P

~~33 A.3 Proofs for Conditional Heteroscedasticity case~~

~~Proof of Theorem 3.2. In order to apply Lemma A.3 we check conditions (i)-(iv) of Lemma A.2 and conditions (a)-(d) of Lemma A.3 for~~

~~1 et′ es Wst = wst T √N and 1 γ′es Vs = vs. √T √N ⊗ (i) Due to serial independence of e conditionally on , we have it F T 2 1 et′ es Σ = E w w′ E . W,T T 2 st st √ |F s=1 t~~

2 et′ es 1 E = tr E(e e′ )E(e e′ ) . √ |F N t t|F s s|F " N # 2 N Recall that e = πf + η . We will use the notation Ω = E (η η′)=dg ω . Then, t t t η t t { i }i=1 2 et′ es 1 E = tr (πE(f f ′ )π′ + Ω )(πE(f f ′ )π′ + Ω ) √ |F N t t |F η s s|F η " N # 1 N = ω4 + ∆ , N i N,T i=1 X where C ∆ E ( f 2 + 1)( f 2 + 1) . N,T ≤ N k tk k sk |F

~~Indeed, ∆N,T has three terms each of which is easy to bound. For example,~~

1 E 1 2 E tr Ωηπ (fsfs′ )π′ max ωi tr (fsfs′ )π′π 1 i N N |F ≤ N ≤ ≤ · |F 1 2 E 2 max ωi max ev(π′π) fs . 1 i N ≤ N ≤ ≤ · · k k |F 2 Since we assumed that max1 i N ωi

~~34 where the last inequality is due to Assumption 5(i). So, we obtain that~~

~~T N 1 1 4 4 Σ (T, N) = lim E [w w′ ] ω = ω Ω = Σ . W,T T 2 st st N i w W s=1 t~~

4 1 N 1 N 1 E η η = E (η η )4 + C E η2 η2 η2 η2 C,  √ it is  N 2 it is N 2 i1t i1s i2t i2s ≤ N i=1 ! i=1 i1=i2 X X X6   N 4 N E 1 1 E 4 1 2 2E 2 2 C πiηis πiηis + C πi1 πi2 ηi1sηi2s ,  √N  ≤ N 2 k k N 2 k k k k ≤ N 2 i=1 i=1 i1=i2 X X X6   4 2 2 where we use Assumption 5(ii,iii), and that πi πi C. Hence, i k k ≤ i k k ≤ 4 P P e e C C E t′ s E f 4 f 4 + E f 4 + E f 4 + C. √ |F ≤ N 2 k tk k sk |F N 2 k tk |F k sk |F " N # Finally, due to Assumption 5(i),

~~e e 4 T 4E W 4 E w t′ s CE w 4( f 4 + 1)( f 4 + 1) < C. k stk ≤ st √ ≤ k stk k tk k sk " N #~~

Thus, condition (ii) of Lemma A.2 holds. (iii) Let us deﬁne a σ-algebra = f , t =1, ..., T . Let us now denote A F∪{ t } 2 2 e e (πfs + ηs)′(πft + ηt) ϑ = E s′ t = E st √ |A N |A " N # " # N 1 2 4 = (f ′π′πf ) + f ′πΩ π′f + f ′πΩ π′f + ω . N s t s η s t η t i i=1 ! X

~~35 We have:~~

~~T T 2 1 es′ et W W ′ Σ = w w′ ϑ st st − W,T T 2 st st √ − st s=1 t~~

~~E so, it is enough to prove convergence of each term separately. Now, [tr(A1A1′ )] is equal to~~

T 2 2 1 es′ 1 et1 es′ 2 et2 E tr(w w′ w w′ ) ϑ ϑ . T 4 s1t1 s1t1 s2t2 s2t2 √ − s1t1 √ − s2t2 s1,s2=1 t1,t2 " N ! N !# X X Notice that in order for a summand from the last sum to be non-zero we need that some indexes in the set s ,s , t , t coincide, and we obtain at most CT 3 non-zero summands. { 1 2 1 2} Each non-zero summand is bounded above by a constant due to the moment assumptions formulated in Assumption 5(i,iii). Thus, E [tr(A A′ )] 0. 1 1 → Notice that due to Assumption 5, and similar to the argument above,

~~N 1 4 C 4 ϑst ωi ( fs + ft + 1) . (A.5) − N ≤ N k k k k i=1 X~~

~~Thus, N T 1 4 1 A = ω w w′ E (w w′ ) 2 N i T 2 st st − st st i=1 ! s=1 t~~

4 1 T E W ′ W W ′ W = E w′ w w′ w E(e′ e e′ e e′ e e′ e ) , s1t2 s2t1 s2t2 s1t1 N 2 s1t2 s2t1 s2t2 s1,t1 s1 t1 t1 s2 s2 t2 t2 s1 |F 36 where we used that the scalar products et′ es = es′ et are scalars and they can be reshuﬄed to make two same-index e stand back to back. Let us bound the N N matrix E(e e′ )= t × t t|F πE(f f ′ )π′ + Ω : t t |F η

E(e′ e e′ e e′ e e′ e ) = tr E(e e′ )E(e e′ )E(e e′ )E(e e′ ) s1 t1 t1 s2 s2 t2 t2 s1 |F t1 t1 |F s2 s2 |F t2 t2 |F s1 s1 |F N max ev E(e e′ ) ≤  t t|F  t s1,s2,t1,t2 ∈{ Y }   N max ev (E(e e′ )) ≤ t t|F t s1,s2,t1,t2 ∈{ Y } NC E f 2 +1 . ≤ k tk |F t s1,s2,t1,t2 ∈{ Y } Also using Assumption 5(i) we obtain that C 4 4 ∗ 8 T E Ws′1t2 Ws2t1 Ws′2t2 Ws1t1 max E wst ft 0. 1 s,t,t∗ T ≤ N ≤ ≤ k k k k → Thus, condition (iv) holds as well. Now we will check assumptions (a)-(d) of Lemma A.3. First, we ﬁnd the limit of the covariance matrix ΣV,T .

N γ′es γ′es ′ γ′π π′γ 1 2 E = E[f f ′ ] + ω γ γ′ √ √ |F √ s s|F √ N i i i N N N N i=1 X Γ′ E (f f ′ )Γ +Γ . → πγ s s|F πγ ω Here we used Assumptions 5(ii,iii). Therefore,

T T 1 γ′es γ′es ′ Σ = var V = E E (v v′ ) V,T s T √ √ |F ⊗ s s s=1 ! " s=1 N N # X X 1 T = E Γ′ f f ′Γ +Γ (v v′ ) T πγ s s πγ ω ⊗ s s s=1 X (Γ′ I )Σ (Γ I )+Γ Ω . → πγ ⊗ kv fv πγ ⊗ kv ω ⊗ v

~~37 The limit matrix is positive deﬁnite since both Γω and Ωv are positive-deﬁnite due to Assumptions 1(i) and 5(iii). Now note that due to Assumption 5(ii),~~

~~4 γ′et 1 4 4 C 4 E = E γ′πf + γ′η CE f + E γ′η , √ |F N 2 k t tk |F ≤ k tk |F N 2 k tk " N #~~

N 4 N N E 4 E 4E 4 2 2 2 2 γ′ηt = γi′ηit γi (ηit)+ C γi1 γi2 ωi1 ωi2 . k k   ≤ k k k k k k i=1 i=1 i1,i2=1 X X X   4 2 Due to Assumptions 2 and 5(iii) we have that E γ′η CN , and thus k tk ≤ γ e 4 E ′ t CE f 4 +1 . √ |F ≤ k tk |F " N #

~~Collecting the pieces,~~

~~1 γ e 4 C T E V 4 CT E E ′ s v 4 E f 4 +1 v 4 0. k sk ≤ T 2 √ |F ⊗k sk ≤ T k sk k sk → " " N # #~~

This gives us the validity of condition (b) of Lemma A.3. 1 N 2 (c) Denote Γ = N − ω γ γ′ Γ . Then, ω,N i=1 i i i → ω T T P 1 γ′et et′ γ VtV ′ ΣV,T = (vtv′) t − T √N √N ⊗ t t=1 t=1 X XT 1 γ′π π′γ E ftf ′ +Γω,N (vtv′) − T √N t √N ⊗ t t=1 TX 1 γ′et et′ γ γ′π π′γ = ftf ′ Γω,N (vtv′) T √N √N − √N t √N − ⊗ t t=1 XT 1 γ′π π′γ γ′π π′γ + f f ′ +Γ E f f ′ +Γ (v v′) T √ t t √ ω,N − √ t t √ ω,N ⊗ t t t=1 N N N N X = A1 + A2.

~~Notice that given the conditional independence of ηit’s, the two terms in the last expres-~~

~~sion, A1 and A2 are uncorrelated, so in order to check condition (c) of Lemma A.3 we can~~

~~38 prove convergence to zero of E A 2 and E A 2 separately. First, k 1k k 2k~~

T 2 E 2 E 1 γ′π ηt′γ γ′ηt π′γ γ′ηt ηt′γ A1 = ft + ft′ + Γω,N (vtvt′) k k  T √N √N √N √N √N √N − ⊗  t=1 X  T 2  1 γ′π ηt′γ γ′ηt π′γ γ′ηt ηt′γ 4 E ft + f ′ + Γω,N vt ≤ T 2 √N √N √N t √N √N √N − k k t=1 " # X 1 E 2 4 C ( ft + 1) vt 0. ≤ T k k k k → The former inequality is due to ηt’s being conditionally serially uncorrelated, and thus the summation over t can be taken outside the expectation of the square; the latter inequality

~~uses bounds on the moments of ηt′γ/√N we derived before. Second, the convergence of~~

~~term A2 is due to Assumptions 5(iv) and 1(iv). Putting all terms together, we obtain that condition (c) is satisﬁed. Finally, we check the condition (d):~~

~~3 es′ 1 γ γ′es2 es′ 1 et es′ 2 et T E W V ′ V W ′ = E w v′ v w′ E . s1t s1 s2 s2t s1t s1 s2 s2t √ √ √ √ |F N N N N ~~

~~Using that scalars could be reshuﬄed to make two same-index et stand back to back and employing conditional independence we obtain:~~

es′ 1 γ γ′es2 es′ 1 et es′ 2 et 1 E = tr γγ′E(e e′ )E(e e′ )E(e e′ ) √ √ √ √ |F N 2 s2 s2 |F t t|F s1 s1 |F N N N N 1 tr( γγ′) max ev E(e e′ ) ≤ N 2 s s|F s s1,s2,t ∈{Y } C E f 2 +1 . ≤ N  k sk |F s s1,s2,t ∈{Y }   1 We use Assumption 2 that N − tr(γγ′) < C and the bound (A.6) we derived before. In

~~the last equality, we also exploit that ft’s are conditionally independent of each other. Thus, Assumption 5 (i) implies that~~

~~3 C T max E Ws1tVs′1 Vs2 Ws′2t 0. 1 t~~

~~39 Proof of Theorem 4.2. We will prove three statements:~~

1 N (i) N − ξ ξ′ Σ ; i=1 V,i V,i → V 1 PN (ii) N − ξ ξ′ Σ ; i=1 W,i W,i → W 1 PN (iii) N − ξ ξ′ 0. i=1 V,i W,i → P (i) Given assumption Γ = 0, we have Σ =Γ Ω . Then, πγ V ω ⊗ v 1 N 1 T T N ξ ξ′ = γ γ′(π′f + η )(π′f + η ) (v v′). (A.7) N V,i V,i NT i i i s is i t it ⊗ s t i=1 t=1 s=1 i=1 ! X X X X After we open up the brackets there will be three diﬀerent types of terms. We will show that T T N 1 p (γ γ′η η ) (v v′) Σ , (A.8) NT i i is it ⊗ s t → V t=1 s=1 i=1 X X X while terms that involve πi′wsπi′wt or ηitπi′fs converge to zero in probability. Indeed,

1 T T N 1 T T N (γ γ′η η ) (v v′) Σ = (γ γ′η η ) (v v′) NT i i is it ⊗ s t − V,T NT i i is it ⊗ s t t=1 s=1 i=1 t=1 s=t i=1 X X X X X6 X T N 1 2 2 + (γ γ′) η v v′ ω E (v v′) . NT i i ⊗ it t t − i t t t=1 i=1 X X We check that the ﬁrst sum in the last expression is negligible:

T T N 2 T T N 1 1 4 4 2 2 E tr (γ γ′η η ) (v v′) γ ω E v v   NT i i is it ⊗ s t  ≤ N 2T 2 k ik i k tk k sk t=1 s=t i=1 ! t=1 s=t i=1 X X6 X X X6 X    N C γ 4ω4 0. ≤ N 2 k ik i → i=1 X

Here we use the conditional cross-sectional and temporal independence of ηit, that is, for 4 s = t we have E(η η η ∗ η ∗ ) = ω if i = j and t, s = t∗,s∗ , and zero otherwise. 6 it is jt js |F i { } { } We also use Assumptions 2 and 5(iii). As for the second sum, we notice that all summands

~~40 in the expression below are uncorrelated with each other, hence~~

N T 2 1 2 tr E γ γ′η (v v′) Σ   NT i i it ⊗ t t − V  i=1 t=1 ! X X   N T  1 2 2 2 = tr E γ γ′η (v v′) E γ γ′η (v v′) N 2T 2 i i it ⊗ t t − i i it ⊗ t t i=1 t=1 X X h i C N T γ 4E v 4 0. ≤ N 2T 2 k ik k tk → i=1 t=1 X X Thus, we showed the convergence (A.8).

~~Now consider terms in (A.7) that involve πi′fsπi′ft:~~

1 T T N γ γ′π′f π′f (v v′) NT i i i s i t ⊗ s t t=1 s=1 i=1 ! X X X 1 N 1 T T = (γ γ′) (π′ π′) I I vec(f f ′) (v v′) . N i i ⊗ i ⊗ i ⊗ kv T kγ ⊗ s t ⊗ s t i=1 ! t=1 s=1 ! X X X Using Assumption 2 and 5(ii) we can show that

~~N N N 1 1 2 2 1 2 (γiγi′) (πi′ πi′) γi πi C πi 0. N ⊗ ⊗ ≤ N k k k k ≤ N k k → i=1 i=1 i=1 X X X~~

Now observe that 2 1 T T E vec(f f ′) (v v′)  T s t ⊗ s t  t=1 s=1 X X F  T T T T  1 E ∗ ∗ ∗ ∗ = tr 2 vec(fsft′) vec(fs ft′ )′ (vsvt′vs vt′ ) T ∗ ∗ ⊗ " t=1 s=1 t =1 s =1 #! X X X X 1 T T C E f 2 f 2 v 2 v 2 < C. ≤ T 2 k tk k sk k sk k tk " t=1 s=1 # X X Here the equality is due to ft’s being serially independent and mean zero conditionally on by Assumption 5(i) and v ; hence, among the four summation indexes at most two F t ∈F may be distinct. The last inequality is due to Assumption 5(i). Thus, we showed that

~~T T N 1 p γ γ′π′f π′f (v v′) 0. NT i i i s i t ⊗ s t → t=1 s=1 i=1 ! X X X~~

~~41 And ﬁnally, we show that~~

~~T T N 1 p γ γ′π′f η (v v′) 0. NT i i i s it ⊗ s t → t=1 s=1 i=1 ! X X X~~

This holds because ηit’s are mean zero, cross-sectionally independent and independent from f conditionally on . This implies that the mean of the sum above is zero, and all t F summands are uncorrelated with each other. The second moment of the sum is bounded above by C T T N γ 4 π 2ω2E f 2 v 2 v 2 0. N 2T 2 k ik k ik i k tk k tk k sk → t=1 s=1 i=1 X X X Thus, we proved statement (i). Let us turn to statement (ii):

~~N N T 1 1 2 2 4 ξ ξ′ Σ = w w′ e e ω N W,i W,i − W,T T 2N st st it is − i i=1 i=1 s=1 t~~

~~As for A , we can notice that all summands with indexes s, t = s∗, t∗ are uncorrelated 1 { } 6 { } with each other, so the correlation for summands with diﬀerent indexes i can come only~~

~~from the πi′ft part. Thus,~~

~~T N 2 E 2 1 E 2 2 4 A1 F = wstwst′ eiteis ωi k k T 4N 2  −  s=1 t~~

~~N N N 2 C 4 4 C 4 2 πi πj max πi πi 0, (A.9) N 2 k k k k ≤ N 2 1 i N k k k k → i=1 j=1 ≤ ≤ i=1 ! X X X~~

42 and hence the term A1 converges to zero. Term E A 2 equals the following expression: k 2kF 1 N T E tr(w w′ ∗ ∗ w w′ ∗ ∗ )e e e ∗ e ∗ e e e ∗ e ∗ . 4 2 s1t1 s1t1 s2t2 s2t2 it1 is1 it1 is1 jt2 js2 jt2 js2 T N ∗ i,j=1 s1,s1, tm

~~Notice that if s < t , s∗ < t∗ and s , t = s∗ , t∗ for m = 1, 2, the only ways m m m m { m m} 6 { m m} when the expectation~~

E(e e e ∗ e ∗ e e e ∗ e ∗ ) = 0 (A.11) it1 is1 it1 is1 jt2 js2 jt2 js2 |F 6 can be non zero is when we place at least four restrictions on the time indexes. Indeed, if s ,s∗, t , t∗ are all distinct, then to get a non zero expectation we need indexes to { 1 1 1 1} coincide as sets: s ,s∗, t , t∗ = s ,s∗, t , t∗ . If the set s ,s∗, t , t∗ contains three { 1 1 1 1} { 2 2 2 2} { 1 1 1 1} distinct indexes, for example, s = s∗ (this is one restriction), then the set s ,s∗, t , t∗ 1 1 { 2 2 2 2} should contain (t1, t1∗) (these are two restrictions), and the remaining indexes should be either equal to each other (one restriction) or equal to the ones previously mentioned (two restrictions). Thus, instead of eight-dimensional summation over time indexes in equation (A.10) we have a four-dimensional summation. Let us consider those terms in (A.10) when the summation index j is equal to i. Notice that since each t index is strictly smaller than the corresponding s index, then any distinct time index can appear in the set s ,s∗, t , t∗,s ,s∗, t , t∗ at most four times, { 1 1 1 1 2 2 2 2} thus any individual error term eit may appear in at most power four. Thus, all non-zero 2 ∗ E 4 E 4 ∗ 4 terms are bounded above by max1 i N,1 s,t,t T wst (ηit)+ C ft < C due ≤ ≤ ≤ ≤ k k k k 4 to Assumption 5(i,iii). There are at most CT N of such terms while the normalization is 2 4 N − T − , hence that sum converges to zero.

Now consider those terms in (A.10) when i = j. Since e = π′f + η , with η ’s 6 it i t it it independent of each other both cross-sectionally and temporally, i = j and s , t = 6 { m m} 6 s∗ , t∗ , we have that all terms including η are zero, and only a non-trivial part of the { m m} it term in (A.11) is the one including πi′ft in place of eit. So, every non-zeros term in the sum (A.10) is bounded by π 4 π 4E w 4 f 8 . So, the sum in (A.10) over j = i is k ik k jk k stk k tk 6 bounded above in the same manner as stated in equation (A.9). Thus, we showed that p p A 0. The convergence A 0 comes from Assumption 1(iii). This ﬁnishes the proof 2 → 3 → of (ii).

~~43 Finally, let us prove statement (iii):~~

~~N T T 1 p (γ v ∗ )w′ e ∗ e e 0. NT 3/2 i ⊗ s st is it is → i=1 s=1 t~~

T 1 3 3 2 2 2 2 2 γi γj πi πj max E vs∗ wst fs∗ fs ft N 2 k kk kk k k k 1 s,t,s∗ T k k k k k k k k k k i,j=1 ≤ ≤ X N 2 N 1 3 1 2 4 2 C γi πi max γi max πi πi 0. ≤ N k kk k ≤ N 2 1 i N k k 1 i N k k k k → i=1 ! ≤ ≤ ≤ ≤ i=1 X X This ends the proof of Theorem 4.2.

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