# Question: The file S12 77 xlsx contains monthly time series data on corporate

The file S12_77.xlsx contains monthly time series data on corporate bond yields. These are averages of daily figures, and each is expressed as an annual rate.

The variables are:

• Yield AAA: average yield on AAA bonds

• Yield BAA: average yield on BAA bonds

If you examine either Yield variable, you will notice that the autocorrelations of the series are not only large for many lags, but that the lag 1 autocorrelation of the differences is significant. This is very common. It means that the series is not a random walk and that it is probably possible to provide a better forecast than the naive forecast from the random walk model. Here is the idea. The large lag 1 autocorrelation of the differences means that the differences are related to the first lag of the differences. This relationship can be estimated by creating the difference variable and a lag of it, then regressing the former on the latter, and finally using this information to forecast the original Yield variable.

a. Verify that the autocorrelations are as described, and form the difference variable and the first lag of it. Call these DYield and L1DYield (where D means difference and L1 means first lag).

b. Run a regression with DYield as the dependent variable and L1DYield as the single explanatory variable. In terms of the original variable Yield, this equation can be written as

Yieldt = Yieldt-1 = a + b(Yieldt-1 – Yieldt-2)

Solving for Yieldt is equivalent to the following equation that can be used for forecasting:

Yieldt = a + (1 + b)Yieldt-1 – bYieldt-2

Try it—that is, try forecasting the next month from the known last two months’ values. How might you forecast values two or three months from the last observed month?

c. The autocorrelation structure led us to the equation in part b. That is, the autocorrelations of the original series took a long time to die down, so we looked at the autocorrelations of the differences, and the large spike at lag 1 led to regressing DYield on L1DYield. In turn, this ultimately led to an equation for Yieldt in terms of its first two lags.

Now see what you would have obtained if you had tried regressing Yieldt on its first two lags in the first place—that is, if you had used regression to estimate the equation

Yieldt = a + b1Yieldt-1 + b2Yieldt-2

When you use multiple regression to estimate this equation, do you get the same equation as in part b?

The variables are:

• Yield AAA: average yield on AAA bonds

• Yield BAA: average yield on BAA bonds

If you examine either Yield variable, you will notice that the autocorrelations of the series are not only large for many lags, but that the lag 1 autocorrelation of the differences is significant. This is very common. It means that the series is not a random walk and that it is probably possible to provide a better forecast than the naive forecast from the random walk model. Here is the idea. The large lag 1 autocorrelation of the differences means that the differences are related to the first lag of the differences. This relationship can be estimated by creating the difference variable and a lag of it, then regressing the former on the latter, and finally using this information to forecast the original Yield variable.

a. Verify that the autocorrelations are as described, and form the difference variable and the first lag of it. Call these DYield and L1DYield (where D means difference and L1 means first lag).

b. Run a regression with DYield as the dependent variable and L1DYield as the single explanatory variable. In terms of the original variable Yield, this equation can be written as

Yieldt = Yieldt-1 = a + b(Yieldt-1 – Yieldt-2)

Solving for Yieldt is equivalent to the following equation that can be used for forecasting:

Yieldt = a + (1 + b)Yieldt-1 – bYieldt-2

Try it—that is, try forecasting the next month from the known last two months’ values. How might you forecast values two or three months from the last observed month?

c. The autocorrelation structure led us to the equation in part b. That is, the autocorrelations of the original series took a long time to die down, so we looked at the autocorrelations of the differences, and the large spike at lag 1 led to regressing DYield on L1DYield. In turn, this ultimately led to an equation for Yieldt in terms of its first two lags.

Now see what you would have obtained if you had tried regressing Yieldt on its first two lags in the first place—that is, if you had used regression to estimate the equation

Yieldt = a + b1Yieldt-1 + b2Yieldt-2

When you use multiple regression to estimate this equation, do you get the same equation as in part b?

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