Macro and Financial Time Series Econometrics

Macro and Financial Time Series Econometrics

Final Exam Prep

 

Instructions:  This exam contains seven questions.  You must complete all questions.  Each question is comprised of sub-questions and the point allocation is provided for reference.  Explain your answers fully using complete sentences and show your work where applicable.  Partial credit will be provided where possible.  Unsupported answers with little or no explanation will receive no credit.  Think before you start writing and Good Luck!

 

  1. (6 points) Suppose you are interested in estimating the AR(1) model, , with a given sample of time series data: . You can assume throughout that the AR(1) is stationary.  You plan to use the standard OLS estimator for the persistence parameter, .

 

  1. Is the OLS estimator of the persistence parameter unbiased? Define exactly what you mean by “unbiased”. (2 points)

 

The OLS estimator is biased on the sense that the expected value of the OLS estimator is not equal to the population parameter.  In terms of an equation we have that:

 

 

  1. What happens to this bias as the sample size grows large? In particular, what happens in the limit as the sample size tends towards infinity? (2 points)

 

As the sample size, T, grows large the bias converges to zero as the OLS estimator is consistent.  In fact, a well-known approximation to the bias of the OLS estimator is given by:

 

 

 

and so one can see that the bias diminishes to zero as T rises.

 

  1. What happens to this bias (in absolute value) as the persistence parameter increases? In particular, suppose you have a sample of T=100 observations.  In which case would you expect the bias to be larger: the case in which the true value of  is 0.7 or 0.9? (2 points)

 

The bias is larger the larger the value of the autoregressive parameter, .  In the equation above we can see that the bias increases (the difference between the expected value of the OLS estimator and the true value gets larger in absolute value) as the autoregressive parameter increases.

 

  1. (6 points) Suppose you are interested in determining whether a sample of data from a time series is consistent with a random walk.
    1. Define the meaning of “random walk” precisely. Use an equation in your answer. (2 points)

 

A random walk is a highly persistent time series in which next period’s value is equal to this period’s value plus a random shock.  The standard equation for a random walk is given by:

 

 

[some students wrote down the equation for a random walk + drift model which was fine too]

 

  1. What is the variance ratio test and how can it be used to test whether a series’ behavior is consistent with a random walk. What is the null hypothesis of the test?  Exactly what property of a random walk is being examined by the test? (2 points)

 

The variance ratio test is a test that compares the variance of the k-period difference of the series with the variance of the 1-period difference as follows:

 

 

Under the null hypothesis the value of the VR test is 1 since the variance of a random walk grows linearly with the horizon, k.  The test is examining whether or not the variance of the series grows linearly with the horizon, k.

 

  1. Suppose you conduct the test with a sample of data and find that the test suggests the series is not a random walk. Can you conclude that the series is also not consistent with a unit root process?  Why or why not?   (2 points)

 

No.  The set of unit root processes is much larger than the set of random walk processes and so you might reasonably find that a series is not a random walk yet it may still be a unit root process.

 

  1. (6 points) Describe the “spurious regression” problem in econometrics. Specifically:
    1. What is the problem and under what conditions is it likely to occur? Why does the problem arise in the first place? (2 points)

 

The spurious regression problem arises when two series, say, y and x, which are highly autocorrelated/persistent are included in a regression.  The problem is that if the underlying series exhibit behavior that is very close to a random walk\unit rot then the usual OLS estimate and associated t-statistics can be very misleading.  This happens because if the two series were actually random walks standard OLS theory does not apply since both the dependent abnd independent variables would have infinite variance and standard OLS theory requires finite variances.  Granger and Newbold showed in an experimental (simulation) setting that two in dependent random walks will exhibit t-statistics signaling a “statistically significant” relationship 77% of the time even though there is no relationship between y and x.    

 

  1. When the problem occurs what happens? What is a researcher likely to find in a sample of data? (2 points)

 

The usual t-statistics used for hypothesis testing are incorrect/inappropriate and one is likely to find “evidence” of a significant relationship when in fact none exists.

  

  1. What measures can be taken to deal with or “fix” the problem? (2 points)

 

One can deal with this problem by regressing the difference in y onto the difference in x since if the model is correctly specified in levels, i.e. the level y is linearly related to the level of x then so too are the first differences.  But, unlike, the levels, the first difference of y and the first difference in x are much more likely to be stationary in which case standard OLS theory supporting the use of the t-statistic will apply.  

 

  1. (10 points) Suppose you have as sample of size T for a time series of interest, , and a two-step ahead forecast of that time series, , as well as an additional variable, , which you suspect may also be useful for forecasting the time series.

 

  1. What does it mean for the forecast to be “rational”? (2 points)

 

The forecast is “rational” if the forecast represents an unbiased measure of the conditional expectation.  In terms of an equation the forecast is rational if the population parameters of the following regression model:

 

   are 0 and 1 respectively.

 

  1. How would you test whether the forecast is “rational” using the data described above. Exactly what test would you perform and what would be your decision rule for deciding whether or not the forecast is rational?   (2 points)

 

I would estimate the regression above and conduct a joint hypothesis test (Wald or F-test) that the population parameters of  and  are 0 and 1, respectively.  If the test is rejected then the forecast is not rational.  If the test is not rejected then the data are consistent with the forecast being rational.

 

  1. What does it mean for the forecast to be “efficient” with respect to z? (2 points)

 

The forecast is efficient with respect to z if z has no predictive power beyond that contained in the forecast.  Once we have accounted for the information in the forecast, z has no forecasting power.

 

  1. How would you test whether or not the forecast is efficient with respect to z? Exactly what test would you perform and what would be your decision rule for deciding whether or not the forecast was efficient with respect to z? (2 points)

 

One way of testing for forecast efficiency is to estimate the following regression:

 

 

 

and test whether or not .

 

  1. Is it possible for a forecast to be rational but not efficient? Is it possible for a forecast to be efficient but not rational? (2 points)

 

Yes.  In the equation above we could easily have that  and  and  in which case the forecast is rational but not efficient and so maybe the forecast is missing some of the information content in z but is otherwise a good forecast.  Likewise, we can have a case in which, say,  but  so that the forecast fully utilizes the information in z but is otherwise mis-specified resulting in irrationality.

 

 

  1. (10 points) Consider the following two equations below:

 

 

  1. Write down this system as a VAR: . Specifically what is  ? (2 points)

 

The VAR system is given by:

 

 

and we have that:

 

 

 

 

 

  1. Write down a matrix equation for the forecast of and given data on  and .  (2 points)

 

The forecast for  and is given by

 

 

  1. What is the relationship between the VAR forecast for and the forecast that would result if one simply ignored  and modeled  as an AR(1)? (2 points)

 

They are identical.  Since Z only depends on its lag and not a lag of x the VAR forecast for  is identical to the AR forecast, .  This can be seen directly by multiplying the above matrix equation out explicitly.  If we do this once we find that:

 

and if we do the third multiplication we get

 

  

 

and note that evaluating the lower row of the matrix yields 0.36*0.6 which is identical to the case of the AR(1).

 

  1. Suppose that you had data on x and z but you did not know the true coefficients provided above. How could you estimate the VAR? (2 points)

 

You could estimate the VAR by MLE which would be accomplished by executing 2 OLS regressions:

 

  1. Regress x onto a constant, a lag of x and a lag of z
  2. Regress z onto a constant, a lag of x and a lag of z

 

  1. Does the VAR model place any restriction on the covariance/correlation between and  ?  If not, why not?  If so, what is the nature of the restriction? (2 points)

 

No it does not.  In a VAR there can be any sort of correlation between   and  . [A few students suggested that there was a restriction in the sense that the correlation between   and   observed at different times (period t and period s) should not be correlated.  This is not a requirement of a VAR model but is an often maintained assumption and so I also accepted this answer.]

 

  1. (10 points) Suppose you are interested in the relationship between two time-series: y and z.  Further, suppose that you suspect that the two series are cointegrated.

 

  1. What exactly does it mean for the series to be cointegrated? (2 points)

 

The two series are cointegrated if:

 

  1. y contains a unit root
  2. z contains a unit root
  3. a linear combination of y and z: does not contain a unit root.

 

  1. Why is cointegration important from a forecasting perspective, if at all? (2 points)

 

Cointegration is important from a forecasting perspective in the sense that there is no long run predictability in either y or z but there is long run predictability in  so while one can’t be very sure about where either series is headed in the long run, one can say where y is headed relative to z.

 

  1. Suppose you suspect the two series to be cointegrated and suppose further you have reason to believe that it is y-z that is “cointegrated”. Describe how you would test whether or not the two series are cointegrated. (2 points)

 

Conduct the following three step test.

 

  1. Test whether y contains a unit root with an ADF test.
  2. Test whether z contains a unit root with an ADF test.
  3. Test whether y-z contains a unit root with an ADF test.

 

There is evidence of cointegration if you do not reject the null of a unit root in 1 and 2 but you do reject the null of a unit root in 3.

 

  1. Now suppose you suspect the series to be cointegrated but you do not have a strong view on the nature of the cointegrating relationship. Now how would you test for cointegration and how does this differ from the test procedure you described in d? (2 points)

 

Do the test above except no introduce a new step 3a to be conducted before step 3

 

3a – estimate the cointegrating relation by executing an OLS regression of y on z andf then compute

 

Also, when conducting the test in step 3 you must use a different statistical table as the critical values of the test change when the cointegrating relationship is estimated.

 

  1. Now suppose you are interested in estimating the cointegrating relation between y and z, i.e. you are not willing to assume that y-z is the right cointegrating relationship but would like to estimate the model and test whether or not  or not.  What is one method besides OLS that can be used to estimate this relationship?  How do you implement this method and how does it differ from OLS?   Why might one use this method rather than OLS? (2 points)

 

One can use dynamic OLS (DOLS)

 

Dynamic OLS is implemented by regressing y onto z and leads and lags of changes in z as follows:

 

 

This method is often used instead of standard OLS since the distribution of standard OLS is non-normal which makes inference difficult but the (asymptotic) distribution of DOLS is normal which makes inference easier.

 

  1. (10 points) Consider the model: where  is distributed iid N(0,1)

 

  1. Suppose is an ARCH(2) process – write down the exact functional form for ? (2 points)

 

The functional form for the ARCH(2) process is given by:

 

 

  1. Under the assumption in a) above what is ? (2 points)

 

Under the assumptions above the mean of y is still zero.  This is because the ARCH model influences volatility but not the mean of y and so the mean of y is unaffected and remains zero.

 

  1. How could you statistically test whether or not ARCH effects are present in a sample of data on y? What specific test would you use?  What is the distribution of this test? (2 points)

 

One can test for ARCH effects by regressing squared values of y onto lagged values of squared y’s.  The test statistic  TxR2 from that regression has a chi-square distribution with m degrees of freedom  – the number of lags used in the regression.  If TxR2 is larger than the relevant critical value then the data are supportive of ARCH effects.

 

  1. Suppose you are not sure about the exact order of the ARCH model. What is one approach for determining the appropriate order of the ARCH model? (2 points)

 

One could use a standard model selection criteria such as AIC or SIC.

 

  1. Suppose you find strong evidence of ARCH effects in the data and suppose further that you determine that the appropriate order of the ARCH model is very long. What is an alternative volatility model that might be appropriate in this context and why?  Write down the exact functional form for the volatility model. (2 points)

 

If there is evidence of a large number of lags in the ARCH model, such as ARCH (20) then it may be sensible to specify a GARCH model such as the canonical GARCH(1,1) model:

 

 

since the presence of the GARCH term  effectively allows for an infinite number of ARCH terms with geometrically declining weights.