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The first method to test for the day of the week effect is OLS regression. This method has been used by many empirical researchers testing for a day of the week effect.

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I am struggling to understand how the sum of the lagged values of the return equation being added removes autocorrelation from the error term.

Siddharth Gopi
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2 Answers2

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Often not including lagged values of dependent variable or independent variables will induce autocorrelation structure in residuals when these values should have been included.

Analyst
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As @Analyst pointed out the inclusion of lagged dependent variables excludes one source of regression error autocorrelation. The autocorrelation can still be present if the lags of dependent variables are included. Here is the mathematical illustration. Suppose the true model is the following

$$Y_t=\alpha+\beta_0X_t+\beta_1X_{t-1}+u_t,$$

where $Eu_t|(u_{t-1},...,X_t,X_{t-1})=0$, meaning that $u_t$ is not autocorrelated and it does not correlated with the regressors. Suppose you are estimating the model

$$Y_t=\alpha+\beta_0X_t+v_t$$

then

$$EX_tv_t=EX_tu_t+\beta_1EX_tX_{t-1}$$

Now if $EX_tX_{t-1}\neq 0$ then you have the ommited variables problem and the autocorellation is the least of your worries, since the OLS estimates in this case are inconsistent. Now if $X_t$ is not autocorrelated then $EX_tv_t=0$ and OLS estimates are consistent and asymptotically normal (if $Eu_t^2<\infty$ and $EX_t^2<\infty$). But \begin{align*} Ev_tv_{t-1}&=E(u_t+\beta_1X_{t-1})(u_{t-1}+\beta_1X_{t-2})\\ &=Eu_tu_{t-1}+\beta_1Eu_{t-1}X_{t-1}+\beta_1Eu_tX_{t-2}+\beta_1EX_{t-1}X_{t-2}\\ &=\beta_1Eu_tX_{t-2} \end{align*} and this might be non zero giving the autocorrelation problem.

So to sum up the claim in the citation is not entirely correct. If lags are omitted this can lead to omitted variable bias and that is the first reason to include them.

mpiktas
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  • What kind of intuition are you looking for? Autocorrelation of variable $u_t$ is defined as $Eu_tu_{t-s}\neq0$ for $s>-1$. So if the claim is that the autocorrelation is removed by adding something, you must calculate $Eu_tu_{t-s}$ for both models, as I did in the answer. Note that the presence or absence of autocorrelation depends on what assumptions are put on the error term. – mpiktas Oct 07 '13 at 12:36
  • We have $v_t=u_t+\beta_1X_{t-1}$, since the true model has $X_{t-1}$, so if we omit it, it is still there. – mpiktas Oct 07 '13 at 12:57
  • Continuing with my previous comment: In this particular case the $X_t$ are the returns, which are usually white noise, so $EX_tX_{t-h}=0$ for all $h=0$. So the possible source of autocorrelation is the correlation between the error term and $X_{t-h}$. If we ignore this correlation OLS is still consistent, but not efficient. If we include these lags we get more efficient estimates by ruling out one possible source of autocorrelation. – mpiktas Oct 07 '13 at 12:59
  • mpiktas, i am struggling to keep but im getting there. i have a related question;

    http://en.wikipedia.org/wiki/Omitted-variable_bias

    in that page, "If the independent variable z is omitted from the regression, then the estimated values of the response parameters of the other independent variables will be given by, by the usual least squares calculation, [Insert Equation]"

    The equation is supposedly for when Z is ommitted, could you tell me what the new equation will be if Z were not ommitted?

     – Siddharth Gopi Oct 07 '13 at 13:52
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    the equation would be $\hat\beta=(\tilde X'\tilde X)^{-1}\tilde X'y$, where $\tilde X=[X,Z]$. – mpiktas Oct 07 '13 at 14:18