This paper documents evidence to investigate if the explanatory variables are always correlated with the error term in the vector autoregression (VAR) model because of the property of the VAR model. I use Christiano <i>et al</i>. (CEE, 2005) as an example to examine this argument empirically. According to the findings of this paper, the impulse responses provided by the structural VAR model may be derived from the biased estimates if we allow variables to be correlated with each other through different horizons. It remains possible for a skeptic to maintain some dominant views inferred from the biased coefficients of the SVAR models.
Since Sims (1980) [
This paper demonstrates a weakness of current empirical practice in the VAR literature: I try to check if the estimated error term is always correlated with the lagged variables on the right-hand side (R.H.S.) of the VAR model because of the structure of the VAR model, which is an analogous question of the correlations between the identified shocks and the explanatory variables in the structural vector autoregression (SVAR) model. If the above correlations exist, it indicates that the estimated coefficients of the VAR model are biased.
Likewise, Friedman (1961) [
Lv (2017) [
The innovation of this paper is that under the new assumption that variables may vary over different horizons, I test if the variables may be correlated with the error terms in the VAR model through a long-horizon perspective. To my knowledge, this is the first paper to check the biased coefficient problem of the VAR model. This paper contributes to the literature by questioning the reliability of the impulses response results derived from the existing SVAR models. When talks about the ceteris paribus, if outside variables will not change, the estimated coefficients of variables in the SVAR models are actually overestimated. I provide convincing evidence that the error term may be always correlated with variables on the R.H.S. of a VAR model through different horizons.
The paper is organized as follows. Section 2 interprets the relations between the identified shocks and the lags of variables on the R.H.S. of the SVAR model. Section 3 uses an existing SVAR model to gauge the biased coefficient problem empirically. Concluding comments are given in Section 4.
The traditional VAR model tries to add sufficient lags to make sure that the equations are not misspecified and the residuals are not autocorrelated. However, if we include more lags in the model, these lags may be correlated with the error term. In this paper, since all variables in the SVAR system are considered as the combinations of shocks, I try to check if the estimated coefficients of the SVAR model are biased.
In detail, a standard SVAR model is usually given by:
y t = ∑ i = 1 p A 0 − 1 A i y t − i + A 0 − 1 u t , u t ∼ N ( 0 , σ 2 ) (1)
where y t is a vector of the model variables, P is the lag length of the variables in the system and u t is the vector of structural shocks. A i denotes the matrices of parameters corresponding to the i th lag and I have left out the vector of constant terms to keep things simple. According to the suggestion in Sims (1980) [
y t = ( 1 + β L + ⋯ + β + ∞ L + ∞ ) A 0 − 1 u t (2)
Equation (2) shows that variables are the combinations of the shocks. We can also transfer Equations (2)-(4) to make the above argument clear.
y t − 1 = ( 1 + β L + ⋯ + β + ∞ L + ∞ ) A 0 − 1 u t − 1 (3)
y t − 1 = ( L + β L 2 + ⋯ + β + ∞ L + ∞ ) A 0 − 1 u t (4)
According to Equations (1)-(4), y t and y t − 1 are both correlated with u t , which implies that the lags of the variables on the R.H.S. of the SVAR model may be correlated with the error term through longer horizons and motivates us to wonder how reliable its impulse response results are.
Likewise, Lv (2017a) [
GDP t = ( L 4 + δ L 5 + ⋯ + δ + ∞ L + ∞ ) ϵ t (5)
It is possible that the shocks selected one-step ahead in u t may take the contributions of the omitted shocks ϵ t as their own. The traditional IRFs assume that these exogenous structural innovations are independent and identically distributed random variables. The policy intuition behind this paper is that the fluctuations of economic time series may not be from random shocks, these innovations may be correlated with each other or the variables through different horizons.
The above argument can be justified on the empirical viewpoints. The IRF, which is often used in estimating the multi-step response of one variable to an impulse in another variable in a system, has been widely used in many articles. In this section, I use the SVAR model in Christiano et al. (CEE, 2005) [
Christiano, Eichenbaum, and Evans construct a model with a moderate degree of nominal rigidities that prevents a sharp rise in marginal costs, generating inertial inflation and persistent output movements after an expansionary shock to monetary policy.
The form of the CEE model is as follows:
F t = ∑ i = 1 p B i F t − i + C u t , u t ∼ N ( 0 , σ 2 ) (6)
F t contains nine quarterly series. The lag length p of the model is set to 4. The order of variables is the real gross domestic product (GDP), the real consumption (RPCE), the GDP deflator (GDPDEF), real investment (INVEST), the real wage (WAGE), labor productivity (PROD), the federal funds rate (FEDFUNDS), real profits (PROFIT) and the growth rate of M2 (M2).
The matrix C is taken to be lower triangular with ones along the principal diagonal. It implies that the variables except real profits and the M2 growth rate will not respond instantaneously to monetary policy innovations.
All estimates reported in this paper are based on the original dataset from 1965Q3-1995Q2 in CEE (2005) [
I begin by checking if the estimated error term is correlated with the first lag of the real output on the R.H.S. of the CEE model in Equation (7). I regress the real output at time t − 1 on the output shocks and the lags of the output shocks estimated from the CEE model.
GDP t − 1 = α 0 + ∑ i = 0 6 β i u GDP , t − p + ε t (7)
Based on the findings of
Then I check if the first lag of the real consumption listed second in the CEE model is correlated with the real output shocks:
RPCE t − 1 = α 0 + ∑ i = 0 6 β i u GDP , t − p + ε t (8)
Estimate | t value | |||
---|---|---|---|---|
(Intercept) | 0.64 | 9.79 | 0.00 | *** |
−0.06 | −0.52 | 0.60 | ||
0.99 | 8.71 | 0.00 | *** | |
0.09 | 0.81 | 0.42 | ||
−0.07 | −0.61 | 0.54 | ||
−0.22 | −1.95 | 0.05 | . | |
−0.07 | −0.61 | 0.55 | ||
−0.21 | −1.87 | 0.06 | . | |
0.47 13.16 on 7 and 102 DF 5.59e−12 |
Estimate | t value | |||
---|---|---|---|---|
(Intercept) | 0.35 | 8.40 | 0.00 | *** |
−0.02 | −0.31 | 0.75 | ||
0.25 | 3.34 | 0.001 | ** | |
−0.04 | −0.58 | 0.56 | ||
−0.08 | −1.05 | 0.30 | ||
0.07 | 0.90 | 0.37 | ||
−0.18 | −2.42 | 0.02 | * | |
−0.12 | −1.57 | 0.12 | ||
0.19 3.39 on 7 and 102 DF 0.003 |
system through different horizons and the estimated coefficients may be biased.
In Equation (9), I demonstrate the impact of the output shocks on the first lag of each variable on the R.H.S. of the CEE model.
Y t − 1 = α 0 + ∑ i = 0 6 β i u GDP , t − p + ε t (9)
According to the outcomes of
Since I find that the first lag of the GDP deflator and M2 are not correlated with the first sixth lags of the output shocks, then I regress them on the other kinds of shocks in Equation (10), respectively:
Y t − 1 = α 0 + ∑ i = 1 2 β GDP , i u GDP , t − p + ⋯ + ∑ i = 1 2 β M 2 , i u M 2 , t − p + ε t (10)
p-value | |||||||
---|---|---|---|---|---|---|---|
*** | . | . | 0.47 | 5.59e−12 | |||
** | * | 0.19 | 2.70e−03 | ||||
0.03 | 0.86 | ||||||
*** | 0.28 | 1.81e−05 | |||||
* | . | 0.12 | 0.06 | ||||
*** | * | * | * | 0.31 | 2.08e−06 | ||
* | *** | . | 0.22 | 5.74e−04 | |||
*** | ** | 0.20 | 1.27e−03 | ||||
0.02 | 0.95 |
Estimate | t value | |||
---|---|---|---|---|
(Intercept) | −0.002 | −0.08 | 0.94 | |
0.007 | 0.18 | 0.86 | ||
−0.19 | −2.75 | 0.007 | ** | |
0.99 | 13.05 | 0.00 | *** | |
0.0002 | 0.01 | 0.99 | ||
−0.005 | −0.05 | 0.96 | ||
0.005 | 0.10 | 0.92 | ||
0.02 | 0.72 | 0.47 | ||
0.0004 | 0.07 | 0.94 | ||
−0.009 | −0.17 | 0.86 | ||
0.09 | 2.20 | 0.03 | * | |
0.18 | 2.61 | 0.01 | * | |
−0.62 | −8.20 | 0.00 | *** | |
−0.04 | −1.83 | 0.07 | . | |
0.29 | 3.25 | 0.002 | ** | |
−0.05 | −0.90 | 0.37 | ||
0.06 | 2.03 | 0.05 | * | |
0.009 | 1.45 | 0.15 | ||
0.06 | 1.13 | 0.26 | ||
0.75 15.89 on 7 and 102 DF <2.20e−16 |
the R 2 for this regression should be approximately zero. The coefficients of all shocks except u PROD , u PROFIT and u M 2 are significant. The estimated coefficients of u GDP , t − 2 u RPCE , t − 2 and u GDPDEF , t − 2 are significant, indicating that the GDP deflator is correlated with the sum of these shocks through different horizons, so the estimated coefficients of the GDP deflator equation in the CEE model may be biased.
The interesting part is that the estimated coefficients of u GDP , t − 1 is not significant, which may imply that the output shock does not affect the GDP deflator contemporaneously as the CEE model assumes or the significance level of the estimated coefficients of u GDP , t − 1 may change if we add more lags into Equation (10).
To sum up, from above evidences, the shocks may be always correlated with the variables in the SVAR model through different horizons. In other words, the
Estimate | t value | |||
---|---|---|---|---|
(Intercept) | 1.78 | 23.71 | 0.00 | *** |
0.07 | 0.52 | 0.60 | ||
0.09 | 0.36 | 0.72 | ||
−0.07 | −0.26 | 0.80 | ||
0.05 | 0.68 | 0.50 | ||
0.13 | 0.42 | 0.68 | ||
0.03 | 0.15 | 0.88 | ||
−0.25 | −2.35 | 0.02 | * | |
−0.002 | −0.08 | 0.94 | ||
1.03 | 5.61 | 0.00 | *** | |
−0.10 | −0.72 | 0.47 | ||
0.23 | 0.95 | 0.34 | ||
−0.29 | −1.10 | 0.27 | ||
−0.03 | −0.44 | 0.66 | ||
−0.17 | −0.55 | 0.58 | ||
0.39 | 2.06 | 0.04 | * | |
−0.32 | −3.05 | 0.003 | ** | |
−0.05 | −2.69 | 0.008 | ** | |
0.53 | 2.95 | 0.004 | ** | |
0.42 3.90 on 7 and 102 DF 6.69e−06 |
error term is more likely to be correlated with the dependent variables and explanatory variables at the same time in the VAR model. Therefore, the impulse response results may be inferred from the biased coefficients of the SVAR model and we should be cautious to interpret these results.
This paper has sought to provide an answer to an ignored question in the literature: does the biased coefficient problem plague the VAR model? I investigate the relationships between the variables and the identified shocks in the SVAR model and find that they are always correlated through different horizons, which is rarely fully persuasive and postulated to be uncorrelated with each other in the conventional models. It implies that the error term is correlated with the explanatory variables on the R.H.S. of the VAR model, which means that its estimated coefficients may be biased. Hence, the biased coefficient problem may limit the credibility of the conclusions drawn from the VAR model. In addition, from the assumption that variables may affect the economy through different horizons, different types of shocks may affect the variables in the structural VAR model over different horizons.
The academic significance of this paper is that my analysis sheds light on the potential problems of the impulse response functions and enhances our understanding on how should we use the VAR model. The distortions of coefficients can be substantial in practice, which invalidates the traditional causal interpretation of the responses of variables to a unit shock and overturns the standard view of how variables affect each other. When talks about the ceteris paribus, if outside variables will not change, the estimated coefficients of variables are actually overestimated. The impulse response results and other conclusions inferred from the biased coefficients of the traditional SVAR models are no means settled issues and needed to be further studied.
The social significance of this paper is that the evidence from the SVAR models may be employed by many center banks to analyze the volatility transmission from a shock to the fluctuations in variables. For example, the oil price may not be the deep factor of recessions according to the instability of its estimated coefficient. It may be just the last straw that breaks the camel. Hence, oil price decreases may help the economy in the short horizon but the real problems which cause recessions still need to be solved.
The limitation of this research is that I only provide the possibility that the estimations of the SVAR model may be biased, but this paper cannot explain how much it will affect the results of the existing literature. For some SVAR models, the biased coefficients may not be important at all because these parsimonious models may capture the main variables which can affect all other variables in the economy. These concerns are beyond the scope of this paper and need to be further studied.
I thank the Editor and the referee for their comments.
Lv, Y.Y. (2017) Does the Biased Coefficient Problem Plague the VAR Model? Theoretical Economics Letters, 7, 454-463. https://doi.org/10.4236/tel.2017.73034