Домашня робота №3

з курсу «Економетрика»

Виконала: студентка ФЕН-3, спеціальності «Фінанси і кредит», групи 2

Нефедова Ольга

Викладач: доцент кафедри фінансів

Подвисоцька Т. О.

Київ — 2014

Home Assignment #4

Assignment #1 (2 points)

State with brief reason whether the following statements are TRUE, FALSE or UNCERTAIN

1) In the presence of heteroscedasticity the usual OLS method always overestimates the standard errors of estimators.

FALSE, typically, but not always, will the variance be overestimated.

2) If multicollinearity is present, the conventional t and F tests are invalid.

TRUE, In the case of multicollinearity, our coefficients are biesed and unefficient (we have large variance). Thus, the confidance intervals tend to be much wider leading to the acceptance of the “zero null hypothesis” more readily. Also t ratio tends to be statistically insignificant.

3) If a regressor that has nonconstant variance is (incorrectly) omitted from a model, the OLS residuals will be heteroscedastic.

FALSE, heteroscedasticity is about the variance of the variance of the error term u_i and not about the variance of the regressor.

4) If a regression model is mis-specified (e.g. an important variable is omitted), the OLS residuals will show a distinct pattern.

TRUE. See answer to 5 below.

5) If residuals estimated from an OLS regression exhibit a systematic pattern, it means heteroscedasticity is present in the data.

FALSE, besides heteroscedasticity such a pattern may result from autocorrelation, model specification errors etc.

6) In the presence of autocorrelation the conventionally computed variances and standard errors of forecast values are inefficient.

TRUE. Since the forecast errors involves , which is incorrectly estimated by the usual OLS formula.

7) The exclusion of an important variable(s) from a regression model may give a significant d-value.

TRUE, a significant Durbin-Watson d could signify specification errors.

8) In the presence of autocorrelation OLS estimators are unbiased as well as inefficient.

TRUE, when autocorrelation is present, OLS estimators are unbiased and they are not efficient.

9) The R² values of two models, one involving regression in the first-difference form and another in the level form, are not directly comparable.

TRUE. We can compare the models which have the same regressand.

10) In case of omitted variable in the regression, the estimates would be still unbiased, but inefficient.

FALSE, if an important variable is omitted the OLS estimates would be biased and inefficient.

Assignment #2 (1 point)

Given a sample of 50 observations and 4 explanatory variables, what can you say about autocorrelation if (a) d=1.05, (b) d= 1.40, (c) d=2.50, (d) d=3.97.

Show your work.

For n=50 and k=4, and , the critical d values are:

d_L = 1,38 4 - d_L = 2,62

d_U = 1,72 4 - d_U = 2,28

(a) positive autocorrelation; (b) inconclusive; (c) inconclusive; (d) negative autocorrelation

When the errors in the regression model have AR (1) serial correlation, why do the OLS standard errors tend to underestimate the sampling variation in the ? Is it always true that the OLS standard errors are too small?

We can reason this from equation 1 (12.4 Wooldridge):

because the usual OLS standard error is an estimate of . When the dependent and independent variables are in level (or log) form, the AR(1) parameter, ρ, tends to be positive in time series regression models. Further, the independent variables tend to be positive correlated, so – which is what generally appears in eq. 1 when the {x_t} do not have zero sample average – tends to be positive for most t and j. With multiple explanatory variables the formulas are more complicated but have similar features.

If ρ < 0, or if the {x_t} is negatively autocorrelated, the second term in the last line of eq. 1 could be negative, in which case the true standard deviation of is actually less than .

Assignment #3 (2 points)

In studying the movement in the production workers’ share in the value added, the following models were considered:

Where Y=labor’s share and t=time. Based on annual data for 1949-1964, the following results were obtained for the primary metal industry:

1) Is there a serial correlation in Model A? In Model B?

There is a serial correlation in Model A, but not in Model B.

2) What accounts for the serial correlation?

The autocorrelation may be due to misspecification of Model A because it excludes the quadratic trend term.

3) How would you distinguish between “pure” autocorrelation and specification bias?

I would need prior knowledge of the probable functional form.

Assignment #4 (2 points)

In a regression of average wages (W) on the number of employees (N) for a random sample of 30 firms, the following regression results were obtained:

1) How do you interpret the two regressions?

As equation1 shows, as N increases by a unit, on average, wages increases by about 0,009 dollars. If we multiply the second equation through by N, we will see that results re quite similar to Eq. (1).

2) What is the author assuming going from Eq. (1) to Eq. (2)? Was he worried about heteroscedasticity? How do you know?

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).

3) Can you relate the slopes and intercepts of the two models?

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).

4) Can you compare the R² values of the two models? Why or why not?

No. The dependent variables in the models are not the same.

Assignment #5 (2 points)

Open the database SLEEP.RAW.

1) Estimate the following regression. Interpret R² , the statistical inference of coefficients and the coefficients themselves.

R²=0.1228, which means that the duration of sleep on 12% is explained by the factors of duration of work, education (years of schooling), age, age^2, number of kids and male and on 78 % is explained by the other factors which we didn’t include in the model.

Coefficients near totwrk and male are statistically significant at 5% and 1% significance level; coefficient near education is significant only at 5 % level; coefficients near age, agesq and yngkid are insignificant.

Interpretation:

B1: each additional minute worked per week decreases the duration of sleep per week by 0,16 minutes.

B2: each additional year of schooling decreases the duration of sleep per week by 11,71 minutes.

B3: each additional year of age decreases the duration of sleep per week week by 8,7 minutes.

B4: agesq increases sleep until some critical value and after it decreases.

B5: if number of kids increases by 1 it will decrease minutes, which observed man sleep at night per week by 0,023 minutes.

B6: duration of man’s sleep per week is 87,75 minutes more than duration of woman’s sleep per week.

b0: in case of zero amount of work, education etc., the amount of minutes, which observed person sleep at night per week is equal to 3840 minutes.

2) Do a visual analysis of heteroscedasticity? What can you say just from the graph?

Just from the graph we can suspect for homoscedasticity, as for all fitted values corresponds approximately the same variance of residuals.

3) Apply full White test for heteroscedasticity (6) in handouts for seminar). What would you conclude?

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