1. Introduction
The effects of common shocks, which may be macroeconomic, technological, institutional, political, environmental, health related, sociological,
etc. (e.g., [
1]), have been recently investigated by various authors, including, among others, [
1,
2,
3,
4,
5,
6]. There are several examples in economics where common shocks may affect the analysis:
Accounting for technological and sociological shocks is extremely important when explaining the healthcare attainments of different countries in terms of, say, their per capita health expenditures and educational attainments (e.g., [
7]).
Financial and political shocks are likely to be relevant when explaining the differences in individual countries’ exchange rate ratios (
i.e., the ratio between purchasing power parity relative to the U.S., say, and the nominal exchange rate relative to the U.S.) in terms of their per capita GDP measured in purchasing power parity—the Balassa-Samuelson hypothesis (e.g., [
8]).
Finance, political, environmental and industry-specific shocks impact the models of executive compensation in which the latter is explained by returns on assets, stock returns, the level of responsibility and gender.
In the cross-country cross-industry analysis of returns to R & D, both global shocks (e.g., the recent financial crisis) and local shocks (e.g., spillovers between a limited group of industries or countries) may be fundamental in explaining output, as well as the explanatory variables (
cf. [
9]).
Other detailed examples (e.g., consumption model and asset pricing model) are provided in Appendix A of [
10].
These shocks induce cross-sectional dependence in panel data models, which is often modelled in a parsimonious way through the use of factors. The earlier contributions allow only for factors in the errors of the model (e.g., [
3,
4]) for which consistent estimation of the parameters of interest could be done by maximum likelihood procedures (e.g., [
11]). Coakley, Fuertes and Smith [
12] suggest an estimation procedure based on principal components applied to the residuals. More recently, it has been noticed by several authors that common shocks would likely affect both the errors and the regressors, or a combination of the two (see among others [
1,
5]), and would thus induce endogeneity requiring more sophisticated estimation procedures. A recent survey can be found in [
13].
Andrews [
1] has studied the conditions for the consistency of the OLS estimator in a cross-section regression with common shocks, which are captured by the sigma algebra generated by the factors. Andrews [
1] assumes that the observed variables are independent and identically distributed given such sigma algebra. He shows that the OLS estimator is consistent if and only if the errors and the regressors are uncorrelated given the sigma algebra generated by unobserved factors and shows that the (possibly re-centred) OLS estimator has an asymptotic mixed normal distribution. However, tests on the coefficients can be constructed using classical distributions under the null if the OLS estimator is consistent. Andrews [
1] also extends his results to the OLS and the fixed effect estimators in panel data with fixed
T.
Recently, Kuersteiner and Prucha [
14] have extended the work of Andrews [
1] by deriving a stable central limit theorem for sample moments under weaker assumptions and have established limiting distributions of GMMand maximum likelihood estimators for general models in which unobservable factors may induce cross-sectional heterogeneity, but do not affect the regressors. The approach of Kuersteiner and Prucha is based on a generalization of Corollary 3.1 of [
15] on martingale difference sequences, which allows them to deal with sequentially exogenous regressors. Kao, Trapani and Urga [
16] also employ Corollary 3.1 of [
15] to investigate panel data models with factor structures.
Although central limit theorems for martingale difference sequences are very powerful tools in time series, their application in cross-sections and panel data with a fixed time dimension is not fully intuitive, and the assumptions employed may be cumbersome. The fundamental reason for this is that in such models, there is no natural order of the observations, and the assumptions must be formulated to guarantee the validity of the derived results for all possible permutations of the sequences of observations.
This paper proposes an alternative approach to that of [
1,
14] to study estimators of linear panel data models in which the errors and the regressors are affected by common shocks represented by common factors. In contrast to the work of [
1,
14], our work employs a conditional strong law of large numbers (e.g., [
17,
18,
19,
20]) and a conditional central limit theorem (e.g., [
19,
20,
21,
22,
23,
24]) from which stable convergence follows.
Conditional strong laws of large numbers and conditional central limit theorems are very similar to their standard counterparts and, therefore, are familiar and intuitive to econometricians. Similarly, the assumptions on which they are based are simple, and one does not have to worry about establishing the validity of the results under all possible permutations of the sequences of observations. We will show in a companion paper that the approach can be used to analyse panel data models with endogeneity due to both simultaneity and factor structures in the errors and the explanatory variables (e.g., [
25]).
The approach that we suggest is based on two steps:
formulation of the assumptions concerning the unobservable and heterogeneous variables conditional on the sigma algebra capturing the common shocks;
application of conditional strong laws of large numbers and conditional central limit theorems to establish the limits of the estimator of interest conditional on the common shocks, from which the unconditional distribution can be obtained.
This approach is used in
Section 2 to study the OLS estimator for the slope coefficients in a panel data model with homogeneous slopes and in
Section 3 to investigate a model with heterogeneous slopes.
Section 4 briefly discusses a fixed effects model, and
Section 5 concludes. All proofs are in the
Appendix.
2. Homogeneous Slopes
We consider a simple panel data model with cross-sectional dependence and correlation between the errors and the regressors:
The observed regressors are split into two groups: those that are not affected by common shocks (e.g., unit characteristics, such as gender, race, age,
etc.),
, and those that may be affected by common shocks,
. The parameters associated with the regressors,
and
, and the constant vector
τ are the same for
. The common shocks are captured by the matrix of unobserved common factors,
, (
cf. [
1]);
and
are factor loadings;
is a purely idiosyncratic random vector with zero mean and arbitrary covariance matrix, which may depend on
i; and
represents the values of the regressors that would be observed in the absence of common shocks. Factors, factor loadings,
and
are not observed. The factor structure generates cross-sectional heterogeneity in the error term of (
1). This also creates correlation between errors
and regressors
. Notice also that the impact of the unobserved shocks is different for each unit depending on the realizations of the factor loadings
and
.
It may help to think of an example where a model like (
1) is applicable. Suppose we are interested in estimating the healthcare attainments of different countries in terms of, say, their per capita health expenditures and educational attainments. In this case,
contains a measure of the educational attainment of country
i over
T years;
is a vector containing a measure of educational attainments over the
T periods for country
i, and
is per capita health expenditure over the same period. There is also a dummy for each time period whose coefficients are in
τ. The common shocks are represented by new procedures, drugs, surgical techniques,
etc. The shocks are not observed by the econometrician. They directly affect the healthcare attainments. However, they also affect health expenditures over time for the different countries. Therefore, the observed health expenditure also includes the common shocks.
All variables are defined on a probability space . The sigma algebra generated by the random vector is denoted by . Notice that is a sub-algebra of . Notice also that expectations and probabilities conditional on are unique up to a.s.equivalence, so that, for example, two conditional expectations that differ only on sets of probability zero are regarded as equivalent. We will regard conditioning on as conditioning on the factors . In the rest of the paper, denotes the Euclidean norm for a vector and the Frobenius norm for a matrix.
We now introduce assumptions on both the observed and the unobserved variables. These are adapted from [
1], but allow for heterogeneity conditional on
. We assume that the matrix of factors, which we do not observe, is random and finite with probability one. Since we regard the time dimension as fixed, no other assumptions for the factors are needed. The following assumptions state that the unobservables are independent sequences of independent random quantities given the factors.
Assumption: Let
δ be a positive constant and Δ be
-measurable and such that
a.s.
- C1
is a sequence of conditionally-independent random vectors given , a.s. and a.s.
- C2
is a sequence of conditionally-independent random matrices given with a.s.
- C3
is a sequence of conditionally-independent random vectors given , a.s., where γ is -measurable, and a.s.
- C4
is a sequence of conditionally-independent random matrices given , a.s., where Γ is -measurable, and a.s.
- C5
, , and are conditionally independent of each other given .
- C6
is uniformly positive definite a.s., where and .
Notice that the expectations in the assumptions hold a.s., since they involve conditional expectations, which are random variables and may fail on sets of probability zero. The random vectors are assumed to be purely idiosyncratic, and the are assumed to form a sequence of independent random vectors given the factors. Since we interpret as a vector of regressors, which would be observed if the common shocks would not affect the regressors, we assume that these form an independent sequence of events that are heterogeneous and may be correlated with . Notice that γ and Γ may be constant or may be functions of the factors.
The factor loadings in both the regressors and the errors are assumed to be independent conditional on
, but not necessarily identically distributed. We will consider a violation of this assumption later. Notice that Assumption C5 only requires independence conditional on
, but does not require the factor loadings in the regressors and errors to be independent unconditionally. Section 7 of [
1,
19,
26] gives a thorough discussion of the relationship between conditional and unconditional independence.
Andrews [
1] considers a similar model for
in which
,
,
and
are conditionally independent of each other given
. This implies that these random vectors and matrices are exchangeable. It is easy to see that exchangeable random variables are identically distributed (but not necessarily independent) unconditionally. On the other hand, Assumption C5 implies that
,
,
and
may be dependent and non-identically distributed unconditionally.
Assumption C6 is needed for the application of the conditional weak law of large numbers. It ensures that the OLS estimator of the slope parameters has an asymptotic non-singular normal distribution conditional on the sigma algebra .
The OLS estimator of the slope parameters is
where
,
,
and
.
Theorem 1. Under Assumptions C1–C6, as , is unbiased, and a.s., where
Theorem 1 shows that, for fixed
T, the estimator
is unbiased and consistent as
N tends to infinity. For the case where the factors affect all of the regressors, unbiasedness is also noticed for the estimator in (
2) by [
27], where the factor loadings in the regressors and the errors are assumed to be mutually independent (which is stronger than Assumption C5).
To obtain the asymptotic distribution of , we need slightly stronger versions of Assumptions C1 and C3 requiring the existence of higher order moments.
Assumption: Let
δ be a positive constant and Δ be
-measurable and such that
a.s.
- N1
is a sequence of conditionally-independent random vectors given . a.s. and a.s. with being -measurable uniformly in i. Moreover, a.s.
- N3
is a sequence of conditionally-independent random vectors given . a.s. and a.s. with being -measurable uniformly in i. Moreover, a.s.
Theorem 2. Under Assumptions N1, C2, N3 and C4–C6, conditional on ,
as where ,
,
Theorem 2 shows that, for fixed T, has a normal asymptotic distribution conditional on . Since the factors are unobservable, one needs to remove the conditioning on them by averaging them out. Thus, the unconditional asymptotic distribution of the OLS estimator of the slope parameters is covariance matrix mixed normal with mixing density given by the density function of the unobserved factors. Notice also that Theorem 2 implies that converges -stably.
We now briefly deal with the problem of hypothesis testing in this set-up. Even if the relevant distribution for
is the unconditional one, which is non-standard, tests of hypotheses can be constructed as usual. In order to do this, we need to be able to “estimate”
and
conditional on
. From the proof of Theorem 1, we know that
For
, we need more restrictive versions of Assumptions C2 and C4 requiring the existence of higher order moments.
Assumption: Let
δ be a positive constant and Δ be
-measurable and such that
a.s.
- CM2
is a sequence of conditionally-independent random matrices given with a.s.
- CM4
is a sequence of conditionally-independent random matrices given with , a.s.
Lemma 1. Given Assumption N1, CM2, N3, CM4 and C5–C6, conditional on ,
as An asymptotic version of the
F-test conditional on
for the null hypothesis that
against the alternative hypothesis
can be easily constructed condition on
where
R is a known and fixed
matrix of rank
and
r is a known and fixed
vector.
We now investigate briefly the effects of dependence between the factor loadings in the regressors and the errors conditional on for the OLS estimator.
Assumption: - D5
, , and are conditionally-independent of each other given .
Assumption D5 differs from C5 because it allows the factor loading in the regressors and those in the errors to be correlated conditional on for each i. This means that the endogeneity induced by the factor structure persists even conditioning on the factors. Therefore, the OLS estimator of the slope parameters will be biased, as shown by the following theorem.
Theorem 3. Under Assumptions C1–C4, D5 and C6, conditional on ,
as where is defined in (3) and .
Notice that by replacing Assumption C5 with Assumption D5, the estimator of has an asymptotic bias conditional on , which depends in a complicated way on the distribution of the factors and of the factor loadings. This implies that unconditionally, the estimator of has a non-degenerate non-standard asymptotic distribution. The intuition behind this result is as follows: since the factor loadings are correlated among themselves, even conditioning on the factors, endogeneity is present even when we condition on .
3. Heterogeneous Slopes
In this section, we consider a more general case, where the coefficients of
and
are allowed to be different for each unit. Precisely, the model is
where
’s are random variables. We are interested in inference about the mean of the unit-specific coefficients
. These parameters are estimated using the OLS estimator defined in (
8) in the previous sections. Some further assumptions are needed.
Assumptions: Let
δ be a positive constant and Δ be
-measurable and such that
a.s.
- H5
, , , and are conditionally-independent of each other given .
- H7
is a sequence of conditionally-independent random matrices given with , a.s.
- NH7
is a sequence of conditionally-independent random matrices given with , and a.s.
Assumption H5 extends Assumption C5 by requiring that is independent of all other variables conditional on .
The next result gives the distributional properties for the OLS estimator for the slope parameters in (
8).
Theorem 4. Let and be as in (3) and (4). As - 1.
Under Assumptions C1–C4, H5, C6 and H7, is unbiased and consistent.
- 2.
Under Assumptions N1, CM2, N3, CM4, H5, C6 and NH7, conditional on where ,
,
and
Theorem 4 shows that for a fixed T, the OLS estimator is unbiased, consistent and asymptotically normal conditional on the factors. This is different from the conditional asymptotic distribution given in Theorem 2 because of the presence of the term . Thus, the effect of random coefficients on the asymptotic properties of the OLS estimator is just an increase in the conditional variance. It also follows from Theorem 4 that converges -stably to a covariance matrix mixed normal random vector. Notice that Theorem 4 reduces to Theorems 1 and 2 if ’s are identically zero.
In order to construct tests on , we need to find a statistic that converges to conditional on as N tends to infinity. This is given in the following lemma.
Lemma 2. Under Assumptions N1, CM2, N3, CM4, H5, C6 and HN7, as ,
conditional on ,
where is given in (6). Tests of hypotheses can then be constructed as outlined in the previous section.