Problem 2.18 - Econometrics (Ch.2 - Basic Ideas of Linear Regression: The Two Va

Question

Problem 2.18 - Econometrics (Ch.2 - Basic Ideas of Linear Regression: The Two Variable Model)

Following Material is for reference for the problem above

Example 2.3. Stock Prices and Interest Rates
Stock prices and interest rates are key economic indicators. Investors in stock markets, individual or institutional, watch very carefully the movements in the interest rates. Since interest rates represent the cost of borrowing money, they have a vast effect on investment and hence on the profitability of a company.
Macroeconomic theory would suggest an inverse relationship between stock prices and interest rates. As a measure of stock prices, let us use the S&P 500 composite index ($1941–1943 = 10), and as a measure of interest rates, let us use the three-month

Treasury bill rate (%). Table 2-6, found on the textbook’s Web site, gives data on these variables for the period 1980–2007.
Plotting these data, we obtain the scattergram as shown in Figure 2-7. The scattergram clearly shows that there is an inverse relationship between the
two variables, as per theory. But the relationship between the two is not linear (i.e., straight line); it more closely resembles Figure 2-5(b). Therefore, let us maintain that the true relationship is:

Note that Eq. (2.23) is a linear regression model, as the parameters in the model are linear. It is, however, nonlinear in the variable X. If you let Z = 1/X, then the model is linear in the parameters as well as the variables Y and Z. Using the EViews statistical package, we estimate Eq. (2.23) by OLS, giving
the following results:

How do we interpret these results? The value of the intercept has no practical economic meaning. The interpretation of the coefficient of (1/X) is rather
tricky. Literally interpreted, it suggests that if the reciprocal of the threemonth Treasury bill rate goes up by one unit, the average value of the
S&P 500 index will go up by about 997 units. This is, however, not a very enlightening interpretation. If you want to measure the rate of change of (mean) Y with respect to X (i.e., the derivative of Y with respect to X), then as footnote 5 shows, this rate of change is given by -B2(1/Xi2), which depends on the value taken by X. Suppose Knowing that the estimated B2 is 996.866, we find the rate of change at this X value as (approx). That is, starting with a Treasury bill rate of about 2 percent, if that rate goes up by one percentage point, on average, the S&P 500 index will decline by about 249 units. Of course, an increase in the Treasury bill rate from 2 percentto 3 percent is a substantial increase.

Interestingly, if you had disregarded Figure 2-5 and had simply fitted the straight line regression to the data in Table 2-6, (found on the textbook’s Web site), you would obtain the following regression:

Here the interpretation of the intercept term is that if the Treasury bill rate were zero, the average value of the S&P index would be about 1229. Again, this may not have any concrete economic meaning. The slope coefficient here suggests that if the Treasury bill rate were to increase by one unit, say, one percentage point, the average value of the S&P index would go down by about 99 units.

Regressions (2.24) and (2.25) bring out the practical problems in choosing an appropriate model for empirical analysis. Which is a better model? How do we
know? What tests do we use to choose between the two models? We will provide answers to these questions as we progress through the book (see Chapter 5). A question to ponder: In Eq. (2.24) the sign of the slope coefficient is positive, whereas in Eq. (2.25) it is negative. Are these findings conflicting?

Table 2-6 - S&P 500 Index and Three-Month Treasury Bill Rate (3-M T Bill) 1980-2007  

Year S&P 3-m T bill (%) 1980 118.78 11.506 1981 128.05 14.029 1982 119.71 10.686 1983 160.41 8.63 1984 160.46 9.58 1985 186.84 7.48 1986 236.34 5.98 1987 286.83 5.82 1988 265.79 6.69 1989 322.84 8.12 1990 334.59 7.51 1991 376.18 5.42 1992 415.74 3.45 1993 451.41 3.02 1994 460.42 4.29 1995 541.72 5.51 1996 670.50 5.02 1997 873.43 5.07 1998 1,085.50 4.81 1999 1,327.33 4.66 2000 1,427.22 5.85 2001 1,194.18 3.45 2002 993.94 1.62 2003 965.23 1.02 2004 1,130.65 1.38 2005 1,207.23 3.16 2006 1,310.46 4.73 2007 1,477.19 4.41 2.18. Refer to Example 2.3, for which the data are as shown in Table 2-6 (on the text- a. Using a statistical package of your choice, confirm the regression results b. For both regressions, get the estimated values of Y (i.e., Y) and compare book's Web site). given in Eq. (2.24) and Eq. (2.25) them with the actual Y values in the sample. Also obtain the residual values, e,. From this can you tell which is a better model, Eq. (2.24) or Eq. (2.25)?

Explanation / Answer

The data is as follows:

The regression between : S&P Index (Y) and T-bill rate (X) produced following results:

The other regression between, Y and (1/X) showed following results.

From the residuals, it can be seen, that the expression given by equation(2.25) is a better one.

Hope this helps.

Year S&P (Y) 3-m T bill (%) (X) (1/ X ) 1980 118.78 11.506 0.086911 1981 128.05 14.029 0.071281 1982 119.71 10.686 0.09358 1983 160.41 8.63 0.115875 1984 160.46 9.58 0.104384 1985 186.84 7.48 0.13369 1986 236.34 5.98 0.167224 1987 286.83 5.82 0.171821 1988 265.79 6.69 0.149477 1989 322.84 8.12 0.123153 1990 334.59 7.51 0.133156 1991 376.18 5.42 0.184502 1992 415.74 3.45 0.289855 1993 451.41 3.02 0.331126 1994 460.42 4.29 0.2331 1995 541.72 5.51 0.181488 1996 670.5 5.02 0.199203 1997 873.43 5.07 0.197239 1998 1,085.50 4.81 0.2079 1999 1,327.33 4.66 0.214592 2000 1,427.22 5.85 0.17094 2001 1,194.18 3.45 0.289855 2002 993.94 1.62 0.617284 2003 965.23 1.02 0.980392 2004 1,130.65 1.38 0.724638 2005 1,207.23 3.16 0.316456 2006 1,310.46 4.73 0.211416 2007 1,477.19 4.41 0.226757

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