当前位置：文档库 › BasicEconometrics

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

(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.

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 l n (w αα)l n (i i u Y w

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

Since the w ’s are constants, collecting terms, we can simplify this model as:

)ln ln αα(ln αA α)ln ln αα(ln 1221*2*21221w w whereA u X u X w w Y i

i i

i i -+=++=++-+=

Comparing this with the second model, you will see that except for the intercept terms, the two models are the same. Hence the estimated slope coefficients in the two models will be the same, the only difference being in the estimated intercepts. (b) The r 2

values of the two models will be the same.

6.11 As it stands, the model is not linear in the parameter. But consider the

following “trick.” First take the ratio of Y to (1-Y) and then take the natural log of the ratio. This transformation will make the model linear in the parameters. That is, run the following regrssion:

i i

i X Y Y 21ββ1ln +=- This model is known as the logit model, which we will discuss in the chapter on qualitative dependent variables.

6.13 (a) For every tenth of a unit increase (0.10) in the Gini coefficient, we would expect to see a 3.32 unit increase in a country ’s sociopolitical instability index. Therefore, as the Gini coefficient gets higher, or a country ’s income inequality gets larger, a country becomes less sociopolitically stable.

(b) To see this difference, simply assess what happens if the Gini coefficient increases by 0.3. So, 33.2 (0.3) = 9.96, indicating an increase of 9.96 in the SPI. (c) Using the standard t test, 8136.28.112.33==t for testing the null hypothesis that the slope coefficient is 0. For 38 degrees of freedom, the critical value from the table in Appendix D is

somewhere between 2.021 and 2.042 (using a two-sided test), so the estimated slope is statistically significant at the 5% level.

(d) Based on the regression results, we can conclude that there is a positive relationship between higher income inequality and greater political instability, although we cannot make a causal statement about the relationship.

CHAPTER 7

7.2 Using the formulas given in the text, the regression results are as follows:

7.12 (a) Rewrite Model B as:

Therefore, the two models are similar.Yes, the intercepts in the models are the same.

(b)The OLS estimates of the slope coefficient of X3 in the two models will be the same.

(c)

(d) No, because the regressands in the two models are different.

7.13 (a) Using OLS, we obtain:

That is, the slope in the regression of savings on income (i.e., the marginal propensity to save) is one minus the slope in the regression of consumption on income. (i.e., the marginal propensity to consume). Put differently, the sum of the two marginal propensities is 1, as it should be in view of the identity that total income is equal to total consumption expenditure and total savings. Incidentally, note that

1^1^βα-=

(b)Yes. The RSS for the consumption function is:

and verify that the two RSS are the same.

(c)No, since the two regressands are not the same.

CHAPTER 8

8.5 (a) Let the coefficient of log K be )1β(ββ32*-+=. Test the null hypothesis that

*β=0, using the usual t test. If there are indeed constant returns to scale, the t value

will be small.

(b) If we define the ratio (Y/K) as the output/capital ratio, a measure of capital productivity, and the ratio (L/K) as the labor capital ratio, then the slope coefficient in this regression gives the mean percent change in capital productivity for a percent change in the labor/capital ratio.

(c) Although the analysis is symmetrical, assuming constant returnsto scale, in this case the slope coefficient gives the mean percent change in labor productivity (Y/L) for a percent change in the capital labor ratio (K/L). What distinguishes developed countriesfrom developing countries is the generally higher capital/labor ratios in such economies.

8.7 Since regression (2) is a restricted form of (1), we can first calculate the F ratio

given in (8..

CHAPTER 9

9.1 (a) If the intercept is present in the model, introduce 11 dummies. If the intercept is suppressed, introduce 12 dummies.

(b) If the intercept is included in the model, introduce 5 dummies, but if the intercept is suppressed (i.e., regression through the origin), introduce 6 dummies. 9.4 The results show that the average price was higher by $5.22 per barrel in 1974 than the other years in the sample. The slope coefficient, $0.30 is the same over the entire sample. The graph will resemble Fig. 9.3 b in the text, with the regression line for 1974 starting at 5.22 on the vertical axis with a slope of 0.30; for the remaining years the regression line will pass through the origin, but with the same slope.

CHAPTER 10

CHAPTER 11

11.2 (a) As equation (1) shows, as N increases by a unit, on average, wages increase by about 0.009 dollars. If you multiply the second equation through by N, you will see that the results are quite similar to Eq. (1).

(b) Apparently, the author was concerned about heteroscedasticity, since he divided the original equation by N. This amounts to assuming that the error variance is proportional to the square of N. Thus the author is using weighted least-squares in estimating Eq. (2).

(c) The intercept coefficient in Eq. (1) is the slope coefficient in Eq. (2) and the slope coefficient in Eq. (1) is the intercept in Eq. (2).

(d) No. The dependent variables in the two models are not the same.

11.6 (a)The assumption made is that the error variance is proportional to the square of GNP, as is described in the postulation. The authors make this assumption by looking at the data over time and observing this relationship.

(b) The results are essentially the same, although the standard errors for two of the coefficients are lower in the second model; this may be taken as empirical justification of the transformation for heteroscedasticity.