# Can you use heteroskedastic time series variables within a regression model?

We are working on a multivariate linear regression model. Our objective is to forecast the quarterly % growth in mortgage loans outstanding.

The independent variables are: 1) Dow Jones level. 2) % change in Dow Jones over past quarter. 3) Case Shiller housing price index. 4) % change in Case Shiller housing price index over past quarter.

In a stepwise regression process, all above variables were selected. And, variables 1) and 3) were surprisingly significant. But, that is when used in combination. Somehow, there is something about the difference in those two indeces that does partly explain the % change in mortgage loans outstanding.

For my part, I find variables 1) and 3) problematic. This is because I believe they are heteroskedastic. And, their respective coefficients confidence interval are therefore unreliable. I also think they may have multicollinearity issues with their related variables based on % change. They may also cause some autocorrelation issues.

However, at first it seems some of my concerns may be overstated. After graphing the residuals of the whole model they seem OK. They don't trend upward. So, it appears the heteroskedasticity is not an issue for the whole model. Multicollinearity is not too bad. The variable with the highest VIF is around 5 much lower than the usual threshold of 10.

Nevertheless, I am still somewhat concerned that even though the whole model seems OK; the specific regression coefficients of the mentioned variables may not be (or more specifically the related confidence intervals).

• Why does heteroscedasticity in the independent variables bother you? Are you using a mixed model in which they are treated as random variables? – whuber Jan 4 '11 at 17:28
• whuber, I am not sure in what direction our model will ultimately go. I don't think we will use our independent variables as random variables. And, if we do I'd be concerned that their respective standard error around the regression coefficient is unstable because of heteroskedasticity. Otherwise, I may be completely wrong to be bothered by heteroscedasticity in the independent variable. But, I don't know if I should or should not be bothered. That's the essence of my question. – Sympa Jan 4 '11 at 18:12
• whuber, giving this more thought here is why an independent level variable bothers me... It is because of its temporal dimension. Let's say you consider what happens to the dependent variable when the Dow is at 5000. Well, a couple of decades ago this may be an economically bullish indicator several thousands points above the relevant level at the time. But, today it has a completely different meaning. It would entail another wrenching crash with dire economic implication. Thus, I don't see how such level variables could really work. – Sympa Jan 4 '11 at 20:38

Your dependent variable is growth. For economic time-series data it is more likely that growth will be a stationary process. This means that it will have constant mean. The level data on the other hand is usually non-stationary. Since your model is linear regression you assume that the true data generating process is

$$Y_t=\alpha_0+X_{1t}\alpha_1+...+X_{kt}\alpha_k+u_t$$

where $u_t$ is a white noise, $Y_t$ is stationary and $X_{kt}$ are non-stationary. Now stationarity implies that

$$EY_t=const=\alpha_0+\alpha_1EX_{1t}+...+\alpha_kEX_{kt}$$.

Now $EX_{kt}$ are functions of time which for non-stationary processes changes with time. So you are implying that

$$\alpha_0+\alpha_1\mu_1(t)+...+\alpha_k\mu_k(t)=const$$

for some non-constant functions $\mu_k(t)$. This places quite severe restrictions on non-stationary processes $X_{kt}$. For example if we have only one independent variable this restriction becomes

$$\alpha_1\mu_1(t)=const-\alpha_0$$

so either $\mu_1$ is constant or $\alpha_1$ is zero. In first case this contradicts the presumption, that $X_1$ is non-stationary in the second case the regression model is of no use.

So in general this is why it is not a good idea to mix levels and growths in regression, unless you are really sure that they are all stationary.

Another problem with time-series regression that for certain class of non-stationarity processes the regression can be spurious. In this case you cannot trust the least squares estimates $\hat{\alpha}_k$, since in spurious regression case their distribution is not normal and does not tend to normal, so usual regression statistics do not apply. For example you can get that $\alpha_k$ is significantly non-zero, when it is actually zero. So before the regression it is always a good idea to test whether your variables are not integrated using some variant of Dickey-Fuller test. I strongly suspect that Dow Jones index is integrated process.

Now as others pointed out, heteroscedasticity in the independent regression variable is harmless. The problems can arise if regression errors are heteroscedastic. Then the least squares estimates will be consistent, but inefficient, also the standard errors should be adjusted, for hypothesis testing.

You are trying to mix level and flow data in one regression. Just notice, that the percent change could be considered as a certain transformation of the level variable (like returns and log-returns in financial econometrics). So since your dependent variable is % change, when you include levels, they try to "mimic" stationarity instead of stochastic movement, and likely may form co-integrating relationship (try to test this anyway). Another "trick" to make level series stationary is to try fractions if they have some sense or consider differences (like interest-rate term-structure).

And I do agree that it is not the heteroscedasticity that forms the problem. Think carefully about the theory you are building your forecasting model on.

Good luck.

• May I ask you to clarify your answer. I can't tell if you think the independent level variables are a problem and should be transformed or not. – Sympa Jan 4 '11 at 19:13
• @Dmitrij Celov, I've upvoted your answer, since I was intending to write something similar, but please fix the language – mpiktas Jan 4 '11 at 20:18
• Even though you try to apply stepwise selection it is not a good practice to pool into one set of the variables those that are in levels (trendy one) and flows (% changes, changes, etc.), especially when you try to model variable that is a % change. My guess is if you do include the lagged level of mortgage loans you could be surprised even more. In this case think about error-correction models if they can help you. Pardons for bad English. – Dmitrij Celov Jan 4 '11 at 20:37
• Dmitrij, intuitively I really agree with you. And, that is why I am asking the question. But, what is the mathematical argument supporting our position? I came up with a reasonably good logic argument as shown in my comments above answering whuber. But, logic is not math... – Sympa Jan 4 '11 at 23:16
• Gaetan, search for the information in Brooks "Introductory Econometrics" or whenever about the "spurious regression", "error correction models". Working with time series non-experimental objects is a bit (but significantly!) different from what you see in general statistics. So you may be concerned about at least several things: are your regression results significant purely at random (an easy check is to perform rolling regression and see the evolution of parameter estimates), are the level parameters more or less the same (then linear restriction? but opposite in sign!). Be cont. – Dmitrij Celov Jan 5 '11 at 7:25

well, to try and help, with heteroskedasticity present, the least squares estimators are still unbiased but no longer most efficient, they no longer have smallest variance violating part of the Gauss Markov theorum. I could be wrong, but I believe you're standard errors are more effected by multicollinearity than heterosked. Multicoll inflates your standard errors, T values become smaller and leading to increased likelihood of retaining false null hypotheses.

• Canyou clarify answer. You start by suggesting that heteroskedasticity is a problem. And, you move on by suggesting that multicollinearity is more of a problem and that heteroskedasticity is Ok. – Sympa Jan 4 '11 at 19:15
• well, without seeing the data I cant tell either way, which is a problem. I was simply stating the actual effects of multicoll and heterosked, because it seems as if in your answer you are concerned with heterosked and the standard errors around the regression plane, but as I understand it, multicollinearity is more of a problem when concerning the standard errors and when estimating coefficients, not heterosked. I'm just unsure of which one actually might be the problem. – JW2 Jan 4 '11 at 19:41
• I think heteroskedasticity is a problem regarding the standard error and the related confidence intervals around a specific regression coefficient. – Sympa Jan 4 '11 at 20:33