Эконометрика — МИЭФ, 2025 midterm 2

A partial-adjustment model is

$$y_t-y_{t-1} = \delta(y_t^*-y_{t-1}),$$

where

$$y_t^*=\beta x_t+u_t.$$

The estimates are $d=0.4$ for $\delta$ and $b=0.8$ for $\beta$. The estimated short-run effect is:

1. $0.16$.

2. $0.32$.

3. $0.4$.

4. $0.64$.

5. $0.8$.

The following information is available for two Logit specifications:

1. $y$ on $x_1$ and $x_2$, with a constant;
2. $y$ on $x_1$ only, with a constant.

What is McFadden $R^2$ for model (2)?

1. $0.64$.

2. $0.72$.

3. $0.75$.

4. $0.778$.

5. $0.889$.

Which is a typical property of maximum-likelihood estimators?

1. Biased but consistent.

2. Unbiased but inefficient.

3. Consistent and asymptotically normal.

4. Always efficient in small samples.

5. Always unbiased.

Consider the ADL(1,1) model

$$Y_t=\alpha+\beta_0X_t+\beta_1X_{t-1}+\rho Y_{t-1}+u_t.$$

What does

$$\frac{\beta_0+\beta_1}{1-\rho}$$

represent?

1. The short-run effect of a temporary increase in $X_t$.

2. The long-run effect of a permanent increase in $X_t$.

3. The coefficient on first differences of $X_t$.

4. The short-run multiplier on $Y_{t-1}$.

5. The error correction term.

Which processes are at least asymptotically stationary?

I. AR(1):

$$X_t=\beta_1+\beta_2X_{t-1}+\varepsilon_t, \qquad |\beta_2|<1.$$

II. Random walk with an MA(1) error:

$$X_t=X_{t-1}+\varepsilon_t+\mu\varepsilon_{t-1}.$$

III. Quadratic trend:

$$X_t=\alpha+\beta t^2+\varepsilon_t.$$

IV. MA(1) with a constant:

$$X_t=\alpha+\varepsilon_t+\mu\varepsilon_{t-1}.$$

1. I and II only.

2. II, III, and IV only.

3. I, III, and IV only.

4. I and IV only.

5. I, II, and IV only.

In a Logit model, what is the probability of success when $Z=0$?

1. $0$.

2. $0.25$.

3. $0.5$.

4. $1$.

5. It may take different values.

Consider the AR(2) process

$$y_t=\beta_0+\beta_1y_{t-1}+\beta_2y_{t-2}+\varepsilon_t.$$

Which equation should be used to test $y_t$ for stationarity with the augmented Dickey-Fuller test?

$$\Delta y_t=\beta+\theta_1y_{t-1}+\varepsilon_t.$$

$$\Delta y_t=\theta_1y_{t-1}+\varepsilon_t.$$

$$\Delta y_t = \beta+\theta_1y_{t-1}+\theta_2\Delta y_{t-1}+\varepsilon_t.$$

$$\Delta y_t = \beta+\theta_1y_{t-1}+\theta_2\Delta y_{t-1} +\theta_3\Delta y_{t-2}+\varepsilon_t.$$

$$\Delta y_t = \theta_1y_{t-1}+\theta_2\Delta y_{t-1} +\theta_3\Delta y_{t-2}+\varepsilon_t.$$

What is a possible drawback of adding lags to time-series models?

1. A decrease in $R^2$.

2. Multicollinearity.

3. Heteroscedasticity of the error term.

4. Autocorrelation of the error term.

5. None of the above.

Which statement about Logit and Probit models is correct?

1. Estimated coefficients are marginal effects of explanatory variables.

2. Estimated coefficients are mean marginal effects of explanatory variables.

3. Estimated coefficients do not directly represent marginal effects of explanatory variables.

4. Probability calculated at the sample means of the regressors equals the sample mean probability.

5. Probability calculated at $Z=0$ equals the sample mean probability.

A model

$$Y_t=\beta_0+\beta_1X_t+u_t$$

is estimated using 80 observations. The residuals $\widehat u_t$ are regressed on $\widehat u_{t-1}$, $\widehat u_{t-2}$, and $X_t$, yielding $R^2=0.12$. The Breusch-Godfrey statistic is computed for two lags.

What is the conclusion at the 5% level, given

$$\chi_2^2=5.99?$$

1. Reject $H_0$ of no autocorrelation through lag 2.

2. Fail to reject $H_0$ of no autocorrelation through lag 2.

3. $H_0$ may or may not be rejected.

4. The Breusch-Godfrey test can detect only first-order autocorrelation.

5. The test statistic falls in an uncertainty region.

Which is not required for the Durbin-Watson test?

1. Regressors are non-stochastic.

2. The model contains a constant.

3. Only first-order autocorrelation is tested.

4. The error term has zero expected value.

5. A lagged dependent variable is a regressor.

The higher power of the ADF $T(\widehat\beta_2-1)$ test compared with the ADF $t$ test means that the $T(\widehat\beta_2-1)$ test is:

1. Relatively more likely to signal stationarity when the examined series is non-stationary.

2. Relatively less likely to signal stationarity when the examined series is non-stationary.

3. Relatively less likely to signal non-stationarity when the examined series is stationary.

4. Relatively more likely to signal non-stationarity when the examined series is stationary.

5. None of the above.

Part 2. Free-response questions — one session, 2 hours without a break.

Structure your answers according to the structure of the questions. When testing hypotheses, state the critical values, degrees of freedom, and significance level used.

Section A. Answer all questions from this section (original Questions 1-2).

A researcher studies the reading habits of residents of a small Russian town. She interviews 50 respondents and defines

$$Y_i= \begin{cases} 1, & \text{if respondent }i\text{ bought a book during the current year},\\ 0, & \text{otherwise}. \end{cases}$$

She believes that buying books depends on:

• $S_i$: years of full-time schooling;
• $E_i$: average monthly income, in thousands of roubles.

Two models are estimated:

Linear Probability Model

$$\widehat Y_i = 0.015+0.099S_i+0.012E_i,$$

with standard errors

$$(0.094)\qquad(0.024)\qquad(0.005).$$

Logit

$$\widehat Y_i = -4.75+0.521S_i+0.067E_i,$$

with asymptotic standard errors

$$(1.33)\qquad(0.168)\qquad(0.036).$$

(a)

• What is the difference between the estimation methods used for the LPM and Logit models?
• How does the interpretation of the dependent variable differ?
• What are the main differences in interpreting the coefficients of the LPM and Logit models?
• Do the estimates indicate that the explanatory factors are significant?
• Are there differences between the two models?
• Which model's results are more trustworthy, and why?
• Calculate the predicted probability of buying a book from both models for a respondent with $S_i=11$ and $E_i=10$.
• Interpret the results and explain why the two predicted probabilities differ.

(b)

• Calculate and compare the marginal effect of schooling $S_i$ for:
  • a respondent with a high-school diploma, $S_i=11$, and monthly income $E_i=10$;
  • a respondent with a university diploma, $S_i=15$, and monthly income $E_i=20$.
• Give an economic explanation for the results.
• The maximum marginal effect of any factor in a Logit model occurs at index value zero. Determine the education level $S$ at which the marginal effect of schooling is maximised when monthly income is 10 thousand roubles.
• Suggest a possible explanation for the result.
• Can the same result be obtained using the LPM?

Using a sample of 36 annual observations for one country, a researcher studies:

• tobacco expenditure $x_t$, in national currency;
• a tobacco price index $p_t$.

Standard errors are in parentheses, $t$ denotes time, and $e_t$ denotes residuals. The researcher estimates four Dickey-Fuller equations:

$$\Delta x_t = 5.55-0.033x_{t-1}-0.16\Delta x_{t-1}-0.13t+e_t,$$

with standard errors

$$(6.36)\qquad(0.065)\qquad(0.19)\qquad(0.057), \tag{1}$$ $$\Delta^2x_t = -1.004\Delta x_{t-1}+e_t,$$

with standard error

$$(0.174), \tag{2}$$ $$\Delta p_t = 0.117-0.106p_{t-1}+1.18\Delta p_{t-1}+0.10t+e_t,$$

with standard errors

$$(0.40)\qquad(0.033)\qquad(0.23)\qquad(0.044), \tag{3}$$

and

$$\Delta^2p_t = -0.126\Delta p_{t-1}+e_t,$$

with standard error

$$(0.091). \tag{4}$$

(a) (12 marks)

• Why is it important to test time series for non-stationarity?
• Discuss the consequences of non-stationarity in regression analysis.
• Using equations (1)-(4), conduct the appropriate tests. Clearly state and explain the critical values used.
• Do the results indicate that $x_t$ and $p_t$ are stationary?
• Do the results indicate that $x_t$ and $p_t$ are difference-stationary?
• What does difference stationarity mean?

(b) (13 marks)

• Equations (2) and (4) do not contain a time trend. Show that taking first differences of a series with a trend, for example

$$y_t=\beta_1+\beta_2y_{t-1}+\beta_3t+u_t,$$

causes the trend to disappear.

• What can be concluded from the resulting expression for the transformed disturbance term?
• In the ADF approach, the researcher includes an additional lag $x_{t-2}$ in the representation

$$x_t=\beta_1+\beta_2x_{t-1}+\beta_3x_{t-2}+u_t,$$

and similarly for $p_t$. What are the advantages and disadvantages of adding this lag?

• Derive the appropriate Dickey-Fuller equation from

$$Y_t=\beta_1+\beta_2Y_{t-1}+\beta_3Y_{t-2}+\gamma t+u_t.$$

• Using equations (1) and (3), recover all coefficients in

$$x_t=\beta_1+\beta_2x_{t-1}+\beta_3x_{t-2}+\gamma t+u_t,$$

and

$$p_t=\alpha_1+\alpha_2p_{t-1}+\alpha_3p_{t-2}+\delta t+u_t.$$

• Draw conclusions about the behaviour of tobacco expenditure and prices.
• Why should Dickey-Fuller equations (1) and (3) be used to estimate coefficients of a series of the form

$$Y_t=\beta_1+\beta_2Y_{t-1}+\beta_3Y_{t-2}+\gamma t+u_t$$

instead of estimating that equation directly by OLS?

Section B. Answer only one question from this section (original Question 3 or Question 4).

According to the Lintner model, the target dividend is determined by earnings, while the actual dividend adjusts gradually toward the target:

$$D_t^*=\alpha+\gamma P_t+u_t,$$ $$D_t-D_{t-1} = \lambda(D_t^*-D_{t-1}),$$

where:

• $D_t$ is the actual dividend;
• $P_t$ is profit;
• $D_t^*$ is the target dividend.

The disturbance satisfies

$$E(u_t)=0, \qquad E(u_t^2)=\sigma_u^2,$$

$u_t$ is uncorrelated with $P_t$, and it is not contemporaneously correlated with $D_t$:

$$\operatorname{Cov}(u_t,D_t)=0.$$

(a) (10 marks)

• What is the basic economic idea behind Lintner's model?
• What advantages does it have over a simple regression of actual dividends on profit?
• Describe the model's dynamics.
• Since $D_t^*$ is unobservable, show how the model can be reduced to an ADL(1,0) model.
• What is the purpose of that transformation?

Lintner's model is extended with an additional error-correction mechanism that accounts for changes in profit:

$$D_t^*=\beta_1+\beta_2P_t+u_t,$$ $$D_t-D_{t-1} = \lambda(D_t^*-D_{t-1})+\delta(P_t-P_{t-1}).$$

(b) (8 marks)

• Explain the logic and economic meaning of the extended model compared with the original model.
• Show that the extended model can be reduced to an ADL(1,1) specification.
• How do its short-run and long-run dynamics differ?
• Derive the short-run and long-run profit effects in terms of the original model parameters.

(c) (7 marks) Estimation using data for several US airline companies, with dividends and profits measured in billions of dollars, gives

$$\widehat D_t = -15.0+0.047P_t-0.031P_{t-1}+0.82D_{t-1}.$$

• Reconstruct the extended model by calculating numerical values of its parameters.
• What conclusions can be drawn about dividend-policy adjustment?
• How many times larger is the long-run profit effect than the short-run profit effect?

Consider a Friedman-type model:

$$C_t=\beta_2Y_t^P+u_t,$$ $$Y_t=Y_t^P+Y_t^T,$$ $$Y_t^P-Y_{t-1}^P = \lambda(Y_t-Y_{t-1}^P),$$

where:

• $C_t$ is actual consumption;
• $Y_t$ is actual income;
• $Y_t^P$ is permanent income;
• $Y_t^T$ is transitory income.

Permanent income is a subjective measure of likely medium-run future income, or expected income. The disturbance satisfies

$$E(u_t)=0, \qquad E(u_t^2)=\sigma_u^2,$$

and $u_t$ is uncorrelated with $Y_t$.

(a) (10 marks)

• What is the basic economic idea behind Friedman's model?
• What advantages does it have over the simple regression

$$C_t=\beta_2Y_t+u_t?$$

• Describe how permanent income evolves over time.
• Since $Y_t^P$ is unobservable, show how the model can be represented as a Koyck distribution. Give the key steps without unnecessarily long transformations.

(b) (8 marks)

• Represent the model as an ADL(1,0) model using a Koyck transformation.
• What are the properties of estimates obtained from the resulting ADL(1,0) model?

(c) (7 marks) For a developed country, the estimated model is

$$\widehat C_t = 0.065Y_t+0.91C_{t-1}.$$

• Find and compare the short-run and long-run marginal propensities to consume.
• Explain the comparison.
• What conclusions can be drawn about the dynamic properties of the estimated model?