BasicEconometrics

CHAPTER 2

2.3 A regression model can never be a completely accurate description of reality. Therefore, there is bound to be some difference between the actual values of the regressand and its values estimated from the chosen model. This difference is simply the stochastic error term, whose various forms are discussed in the chapter. The residual is the sample counterpart of the stochastic error term.

2.6 Models (a), (b), (c) and (e) are linear (in the parameter) regression models. If we let α= ln β1, then model (d) is also linear.

2.7 (a) Taking the natural log, we find that ln Yi = β1 + β2 Xi + ui, which becomes a linear regression model.

(b) The following transformation, known as the logit transformation, makes this model a linear regression model:

ln [(1- Yi)/Yi] = β1 + β2 Xi + ui

(c) A linear regression model

(d) A nonlinear regression model

(e) A nonlinear regression model, as β2 is raised to the third power.

2.9 (a) Transforming the model as (1/Yi) = β1 + β2 Xi makes it a linear regression model.

(b) Writing the model as (Xi/Yi) = β1 + β2 Xi makes it a linear regression model.

(c) The transformation ln[(1 - Yi)/Yi] = - β1 - β2 Xi makes it a linear regression model.

Note: Thus the original models are intrinsically linear models.

CHAPTER 5

5.5 (a) Use the t test to test the hypothesis that the true slope coefficient is one.That is obtain:821.00728

.010598.1)

β(1

β2^2^=-=-=se t For 238 df this t value is not significant even at α=10%. The conclusion is that over the sample period, IBM was not a volatile security.

(b) Since 4205.23001.07264.0==t , which is significant at the two percent level of

significance. But it has little economic meaning. Literally interpreted, the intercept value of about 0.73 means

that even if the market portfolio has zero return, the security's return is 0.73 percent.

5.8 (a) There is a positive association in the LFPR in 1972 and 1968, which is not surprising in view of the fact since WW II there has been a steady increase in the LFPR of women.

(b) Use the one-tail t test.

7542.11961

.016560.0-=-=t . For 17 df, the one-tailed t value at α=5% is 1.740.Sincethe estimated t value is significant, at this level of significance, we can reject the hypothesis that the true slope coefficient is 1 or greater.

(c) The mean LFPR is : 0.2033 + 0.6560 (0.58) ≈ 0.5838. To establish a 95% confidence interval for this forecast value, use the formula: 0.5838 ± 2.11(se of the mean forecast value), where 2.11 is the 5% critical t value for 17 df. To get the

standard error of the forecast value, use Eq. (5.10.2). But note that since the authors do not give the mean value of the LFPR of women in 1968, we cannot compute this standard error.

CHAPTER 6

6.2 (a) & (b) In the first equation an intercept term is included. Since the intercept in the first model is not statistically significant, say at the 5% level, it may be dropped from the model.

(c) For each model, a one percentage point increase in the monthly market rate of return lead on average to about 0.76 percentage point increase in the monthly rate of return on Texaco common stock over the sample period.

6.3 (a) Since the model is linear in the parameters, it is a linear regression model. (b) Define Y* = (1/Y) and X* = (1/X) and do an OLS regression of Y* on X*. (c) As X tends to infinity, Y tends to (1/β1).

(d )Perhaps this model may be appropriate to explain low consumption of a commodity when income is large, such as an inferior good.

6.6 We can write the first model as:

*

i 2211)X ln(w αα)ln(i i u Y w ++=, that is *

22211ln αln ααln ln i i i u X w Y w +++=+, using properties of the logarithms.