Regression – Solution for Autocorrelation in Linear Regression Model with Economic Data

autocorrelationmultivariate regressionregression

I am trying to estimate a multivariate linear regression model in the form of:

$Y(t) = c + b_1*X_1(t) + b_2*X_2(t) + b_3*X_3(t) + b_4*X_4(t)$

All my variables (both Xs and Y) are Year on Year changes of economic data measured Quarterly (Frequency = Quarterly).

When I run the regressions, my model suffers from Autocorrelation. This is logical if we think that between 2 consecutive quarters of yearly changes there is an overlap of 3 quarters.

Now, my question is. Would it be acceptable to introduce a lagged version of the dependent variable in my regression?
i.e. estimate a model in the form of:

$Y(t) = c + k*Y(t-1) + b_1*X_1(t) + b_2*X_2(t) + b_3*X_3(t) + b_4*X_4(t)$

When I do that, there is not an Autocorrelation problem anymore. I would like though to make sure that this does not create any other problem that I can't imagine.

Thanks a lot in advance.

Best Answer

That is strange, but not impossible. Maybe tell us how you tested for serial correlation (SC)?

Anyway, I would worry less about SC and more about stationarity of the data. You can always use either FGLS, or robuts standard errors for inference - and these are easy to compute. But if you do not have stationairty, you get inconsistent estimates (unless you make the very strong Gauss Markov or the classical linear model assumptions). And this is by far, a much bigger problem.

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Regression – Estimating $b_1 x_1+b_2 x_2$ Instead of $b_1 x_1+b_2 x_2+b_3x_3$

The issue you need to worry about is called endogeneity. More specifically, it depends on whether $x_3$ is correlated in the population with $x_1$ or $x_2$. If it is, then the associated $b_j$s will be biased. That is because OLS regression methods force the residuals, $u_i$, to be uncorrelated with your covariates, $x_j$s. However, your residuals are composed of some irreducible randomness, $\varepsilon_i$, and the unobserved (but relevant) variable, $x_3$, which by stipulation is correlated with $x_1$ and / or $x_2$. On the other hand, if both $x_1$ and $x_2$ are uncorrelated with $x_3$ in the population, then their $b$s won't be biased by this (they may well be biased by something else, of course). One way econometricians try to deal with this issue is by using instrumental variables.

For the sake of greater clarity, I've written a quick simulation in R that demonstrates the sampling distribution of $b_2$ is unbiased / centered on the true value of $\beta_2$, when it is uncorrelated with $x_3$. In the second run, however, note that $x_3$ is uncorrelated with $x_1$, but not $x_2$. Not coincidentally, $b_1$ is unbiased, but $b_2$ is biased.

library(MASS)                          # you'll need this package below
N     = 100                            # this is how much data we'll use
beta0 = -71                            # these are the true values of the
beta1 = .84                            # parameters
beta2 = .64
beta3 = .34

############## uncorrelated version

b0VectU = vector(length=10000)         # these will store the parameter
b1VectU = vector(length=10000)         # estimates
b2VectU = vector(length=10000)
set.seed(7508)                         # this makes the simulation reproducible

for(i in 1:10000){                     # we'll do this 10k times
  x1 = rnorm(N)
  x2 = rnorm(N)                        # these variables are uncorrelated
  x3 = rnorm(N)
  y  = beta0 + beta1*x1 + beta2*x2 + beta3*x3 + rnorm(100)
  mod = lm(y~x1+x2)                    # note all 3 variables are relevant
                                       # but the model omits x3
  b0VectU[i] = coef(mod)[1]            # here I'm storing the estimates
  b1VectU[i] = coef(mod)[2]
  b2VectU[i] = coef(mod)[3]
}
mean(b0VectU)  # [1] -71.00005         # all 3 of these are centered on the
mean(b1VectU)  # [1] 0.8399306         # the true values / are unbiased
mean(b2VectU)  # [1] 0.6398391         # e.g., .64 = .64

############## correlated version

r23 = .7                               # this will be the correlation in the
b0VectC = vector(length=10000)         # population between x2 & x3
b1VectC = vector(length=10000)
b2VectC = vector(length=10000)
set.seed(2734)

for(i in 1:10000){
  x1 = rnorm(N)
  X  = mvrnorm(N, mu=c(0,0), Sigma=rbind(c(  1, r23),
                                         c(r23,   1)))
  x2 = X[,1]
  x3 = X[,2]                           # x3 is correated w/ x2, but not x1
  y  = beta0 + beta1*x1 + beta2*x2 + beta3*x3 + rnorm(100)
                                       # once again, all 3 variables are relevant
  mod = lm(y~x1+x2)                    # but the model omits x3
  b0VectC[i] = coef(mod)[1]
  b1VectC[i] = coef(mod)[2]            # we store the estimates again
  b2VectC[i] = coef(mod)[3]
}
mean(b0VectC)  # [1] -70.99916         # the 1st 2 are unbiased
mean(b1VectC)  # [1] 0.8409656         # but the sampling dist of x2 is biased
mean(b2VectC)  # [1] 0.8784184         # .88 not equal to .64

Solved – How to regress household income on three factors

The fact that the p-value is 5,5% only means that the coefficient of kl6 is not statistically significant at 5% level -but it is significant at 6% level, and more so at 10% level. The "5% rule" has no scientific justification whatsoever - it has historical justification and perhaps social justification, but that's another matter and a very large discussion.

Interpretation-wise, the negative coefficient gives us the marginal effect of the number of little children on household income after the educational effect has been controlled for (by the existence of the other two regressors). So what does it say? That more little children tend to reduce household income. This may appear counter-intuitive because one could think "more children provide stronger incentives to earn more income in order to provide for the larger family". Yes, but more children also mean greater demands on the parents time that must be devoted to the children, and so less time available to work and earn income. I would suggest to try a regression where you include in addition the kl6 squared. If this squared regressor obtains a negative coefficient and the plain kl6 obtains a positive coefficient, then you are possibly looking at a non-monotonic relation (i.e. that there is an income-maximizing number of little children below or above which income tends to be lower).

PS: "How can I keep a regressor in a regression?" is the mother-question that leads to data-tampering in those ingenious ways only statistics can offer. I would suggest not to ask yourself again such a question. The regression results are what they are. Statistics should not be the brush with which we paint the world in the colors we want.

Best Answer

Related Solutions

Regression – Estimating $b_1 x_1+b_2 x_2$ Instead of $b_1 x_1+b_2 x_2+b_3x_3$

Solved – How to regress household income on three factors

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