Question: When we carry out forecasting, we simply take the estimation...

22 May 2024,1:11 PM

Time Series models

1. When we carry out forecasting,

(a) we simply take the estimation equation we obtained from our regression of the in-sample data, say, an ARDL(3,1), and plug in the values for the x and y variables on the right hand side of the equation to create a predicted value of y (a forecast) in the forecast sample. In this process of computing the forecast (b) we ignore the standard errors from the original ARDL(3,1) regression, but (c) we are able to compute a forecast interval using the root mean squared error (RMSE) of the forecast.

a. The statement is correct.

b. Parts (b) and (c) of the statement are wrong.

c. Part (c) of the statement is wrong.

d. The statement is wrong in all three parts.

 

2. We use tools like the ACF (autocorrelation function) and the related PACF (partial ACF) to learn about the dynamic structure of a variable (e.g. as the name suggests whether it is ‘autocorrelated’) but we cannot use these to judge the dynamic structure of a residual series.

a. This statement is wrong in its entirety.

b. Only the first part of the statement (‘learn about... of a variable’) is wrong.

c. Only the second part of the statement (‘cannot use... residual series’) is wrong.

d. Both parts of the statement are correct.

 

3. Which of the following is a distributed lag (DL) model of order 2?

a. 𝑥𝑡 = 𝜓 + 𝛽𝑥𝑡−1 + 𝛾𝑥𝑡−2 + 𝜀𝑡

b. 𝑦𝑡 = 𝜓 + 𝛼𝑥𝑡 + 𝛽𝑥𝑡−1 + 𝜀𝑡

c. 𝑥𝑡 = 𝜓 + 𝛽𝑦𝑡−1 + 𝛾𝑦𝑡−2 + 𝛿𝑦𝑡−2 + 𝜀𝑡

d. None of the above

 

4. The first difference of a variable series denotes the lagged value at time t-1 subtracted from the contemporaneous value at time t.

a. True

b. False

5. Which formula is used to compute the long-run multiplier for the following dynamic regression model: 𝑥𝑡 = 𝜓𝑦𝑡−1 + 𝜌𝑦𝑡−2 + 𝛼𝑥𝑡−1 + 𝛽𝑥𝑡−2 + 𝜈 + 𝜀𝑡

a. (𝛼 + 𝛽)/(1 − (𝜓 + 𝜌))

b. (𝛼 + 𝛽)/(1 − (𝜓 − 𝜌))

c. (𝜓 + 𝜌)/(1 − (𝛼 − 𝛽))

d. (𝜓 + 𝜌)/(1 − (𝛼 + 𝛽))

 

6. When we carry out a unit root test (e.g. a DF or ADF test), we can determine whether a variable series is stationary or nonstationary. If the test cannot reject the null that the series is nonstationary, we cannot tell whether it is I(1) or I(2)…

a. The statement is entirely wrong.

b. Only the second part of the statement (‘we cannot tell…’) is wrong.

c. … unless we conduct a unit root test of the variable series in first differences (and, if need be, also second differences, etc, until we reject the null).

d. None of the above.

 

7. If two variables are cointegrated, this means that in the long-run they form a stable equilibrium relationship, which we can represent with a constant parameter.

a. The statement is wrong, we do not have constant parameters due to nonstationarity.

b. Since the underlying variables are nonstationary (e.g. random walks), we can only interpret the sign of the cointegrating relationship, but not the magnitude.

c. Only the part of the statement about the ‘equilibrium relationship’ is correct.

d. The statement is correct.

 

8. When a time series regression model contains one or more lagged dependent variable(s) and the residuals are serially correlated then OLS estimates…

a. … of all variables in the model will be biased.

b. … of only the lagged dependent variables in the model will be biased.

c. … of only the lagged dependent variables in the model will be unbiased.

d. … will be unbiased if we use Newey-West standard errors.

 

Limited Dependent Variable models

9. The marginal effects of x are constant across all values of x in a logit or probit model but vary across x in a Linear Probability Model.

a. Only the second part of the statement (‘vary… Linear Probability Model’) is correct.

b. Only the first part of the statement (‘constant… logit or probit model’) is correct.

c. Both parts of the statement are wrong.

d. The statement is correct.

 

10. The -AAA- model by construction suffers from heteroskedastic residuals whereas this is not the case for the -BBB- model. Fill in for -AAA- and -BBB-.

a. -AAA- is logit, -BBB- is linear probability

b. -AAA- is probit, -BBB- is logit

c. -AAA- is logit, -BBB- is probit

d. None of the above.

 

11. Since the probit and logit estimators are for nonlinear equations the ceteris paribus assumption of standard multiple regression no longer applies.

a. False

b. True

 

Panel Data models

12. Below is a generic macro panel regression model for a dependent variable y and an independent variable x. This equation represents… 𝑦𝑖𝑡 = 𝛼𝑖 + 𝛽𝑖𝑥𝑖𝑡 + 𝜆𝑖 ′𝑓𝑡 + 𝜀𝑖𝑡

a. a fixed effects (𝛼𝑖) model with additional year fixed effects 𝜆𝑖 ′𝑓𝑡 .

b. a random effects model.

c. a heterogeneous parameter model with a multifactor error structure.

d. None of the above.

 

13. In a simple pooled panel OLS model we do not capture the individual-level effects and hence we know that by construction any individual-level effects are uncorrelated with our regressors.

a. This statement is correct.

b. Only the first part (‘do not… individual effects’) of the statement is wrong.

c. Only the second part of the statement (‘know that… uncorrelated’) is wrong.

d. None of the above. 

 

14. Below we present some test output from Stata, related to an empirical model of pupil test performance (dependent variable: the share of pupils in a Michigan school district which pass a maths test) and various determinants.

a. We are conducting a Hausman test which suggests the difference between RE and FE results is not systematic.

b. We are conducting a Hausman test which suggests we should prefer the RE model.

c. We are conducting a Hausman test which suggests the RE model is consistent under the null.

d. None of the above.

 

15. Assume panel data for individuals (i) over several years (t). If individual-level information on human capital (e.g. years of schooling) in a wage regression does not vary over time (because individuals in this sample went to school first, then joined the labour force but never returned to education) then we cannot estimate the effect of human capital on wage if we adopt a random effects (RE) model.

a. True

b. False

 

16. If there is a budget constraint, then the wealth and spending of a household cannot permanently diverge. What holds for individual households, should also hold for entire economies. In this question we examine this issue using quarterly data from the United States for the period 1982-2019. Our analysis adopts current asset value (w_curr) as our proxy for wealth alongside current consumption (c_curr). In Figure 16.1 we chart the time series evolution of the two series over the sample period. Figure 16.2 shows scatter plots of the levels of quarterly asset value and wealth (left panel) as well as of the first difference of these variables (right panel). Throughout this question, we adopt a 10% level of statistical significance for any testing procedures

a. In time series econometrics we have a unique methodology for analysing a long-run equilibrium relationship when we may be concerned that the relationship in the data is merely spurious. Discuss the concept underlying this methodology (how can Figure 16.2 speak to this?) and how spurious regression can arise. [7 marks]

b. Table 16.1 reports results from some statistical tests carried out on the two variables of interest, consumption and wealth. Describe the tests and their rationale. What do you conclude for the time series properties of these variables? Can you deduce the order of integration from the test results provided? [8 marks]

c. We adopt the Engle-Granger two-step methodology to analyse whether consumption and wealth exhibit a long-run equilibrium relationship. Describe in detail what this methodology entails, including assumptions about the time series properties of the variable series. [8 marks]

d. What do you make of the results presented in Table 16.2, panel (a)? Next, interpret, with reference to your above discussion of the methodology, the results in Table 16.2, panel (b) – you can adopt a 10% level of statistical significance for your econometric judgement. What do you conclude for the long-run wealth-consumption relationship for the 1983-2019 period? What about the two sub-periods? [12 marks]

 

17. In this question we study the effect of race on market outcomes. In a seminal online experiment conducted in 2009, economists Jennifer Doleac and Luke Stein (2013) advertised new Apple iPods [a digital personal stereo popular before the launch of the iPhone] for sale through 329 local online market platforms across the United States. They included a photograph of a hand holding the iPod for sale in each advert, which enabled them to study the effect of being a black seller (i.e. an advert including a photo of a dark-skinned hand holding the iPod) on the number and magnitude of offers from potential buyers compared with that of a white seller (light-skinned hand). They also studied the effect of being a white seller with a tattoo (light-skinned hand featuring a wrist tattoo): it is hypothesised that tattooed sellers are likely to be discriminated against for many of the same reasons as black sellers. Just to be very clear: there were no black or white or tattooed sellers; all iPods were sold by Doleac and Stein, who randomly varied the photographs accompanying the adverts.

In the following we focus on the response characteristics of potential buyers who got in touch with the seller by email: rather than questions related to the economics of the sale (e.g. how much the potential buyer is offering) we examine issues of underlying levels of respect or trust. The research question for this part of the experiment is: Do black sellers experience different levels of respect or trust?

We investigate the following dependent variables:

Name: the potential buyer included or signed their name in the email;

Polite: the potential buyer was polite, using expressions like ‘please’, ‘thank you’ or variations (‘pls’, ‘thx’) in the email;

Personal: the potential buyer included a personal story, presumably to appeal to the seller’s sentiment and secure a lower price.

The two primary independent variables of interest are:

Black: a dummy variable equal to 1 if the hand shown in the advertisement was dark-skinned and 0 otherwise;

Tattoo: a dummy variable equal to 1 if the hand shown in the advertisement was light-skinned and tattooed and 0 otherwise;

From a host of additional controls we report below the following:

Christmas: a dummy variable equal to 1 if the advert was posted in the lead-up to Christmas and 0 otherwise;

Valentine’s Day: a dummy variable equal to 1 if the advert was posted in the lead up to Valentine’s Day and 0 otherwise;

Median HH income (log): the average household income in the locality in logs of $US 1,000; % population

White: the share of population in the locality who is White; Poverty Rate: the share of the local population living in poverty. 

a. Table 17.1 columns (1) to (3) present results from three different models (one for each of the three dependent variables) estimated using ordinary least squares (OLS). Discuss the marginal effects from all three models, covering statistical significance and economic interpretation. If some results are surprising to you then you should remark on this. What do we learn from these results to answer our research question? [10 marks]

b. Discuss in detail what the advantages and problems of using OLS are instead of Logit or Probit when analysing binary outcomes. In your discussion, touch upon the scatter plots in Figure 17.1, which present predicted outcomes (y-axis, here: the potential buyer uses their name in the email to the seller) plotted against a number of the regressors. Are some advantages or problems seemingly less significant in the current application? [6 marks]

c. Table 17.1 columns (4) to (6) present raw estimates from a probit regression. Interpret the results: what can you say about the statistical and economic interpretation of these three models? [6 marks]

d. Explain how ‘marginal effects’ are constructed for a least squares and a probit estimator in a limited dependent variable model and the conceptual differences between them. Distinguish between marginal effects for continuous and binary independent variables. You can use primarily words but, if you wish, also graphs and equations in your explanations. [6 marks]

e. In Table 17.2 we report a number of diagnostic test results for the OLS and Probit models in columns (1) and (4) of Table 17.1. Which model would you prefer on the basis of overall model success rate (correctly predicted outcomes)? Which would you prefer on the basis of model maximised likelihood? Finally, if you only cared about which of these two models best predicts which potential buyer uses their name in the response email, which one would you prefer? [7 marks]

 

Notes: These scatter plots present the predictions from Model (1) in Table 17.1 (a dummy equal to 1 if the potential buyer used their own name in the email) graphed against the dummy variables ‘Black’ and ‘Tattoo’ as well as the continuous variable ‘Median HH income (logs)’. Each small circle represents the predictions for a single individual

 

Notes: We present estimates (slope coefficients and, in square brackets, absolute tstatistics) from OLS and Probit regressions for three different dependent variables as described in the text. A constant term and a host of other control variables are included in each model but not reported here. The final row of the table indicates how frequently the dependent variable of the respective model is equal to 1. *, ** and *** indicate statistical significance at the 10%, 5% and 1% level.

 

Notes: These are diagnostic results for the models for ‘name’ in columns (1) and (4) of Table 17.1, where the former is for an LPM and the latter for a probit model. Max LL* is the maximised log likelihood

 

18. In this question we study the returns to R&D (research and development) investment in a ‘Griliches knowledge production function’. Griliches’ (1979) work introduced R&D stock as additional input into a standard log-linear Cobb-Douglas production function and this approach has proven very popular: in this setup the coefficient on log R&D stock can be interpreted as the percentage return to R&D investment. A related literature, following the work by Grossman and Helpman (1991) and Coe and Helpman (1995), studies ‘knowledge spillovers’ (e.g. how R&D investment in the transport equipment sector also fosters growth in the machinery and equipment sector). Curiously these two strands of literature for a long time developed in isolation (and ignorance) of each other. More recently, researchers have asked whether we can ignore knowledge spillovers when estimating the returns to R&D, and it is this question we investigate in the following. Our sample is made up of twelve manufacturing sectors (from ‘Food and Beverages’ to ‘Transport Equipment’ production) in ten advanced OECD countries over the 1980 to 2005 period. The unit of analysis is the country-sector: to avoid confusion we use the single subscript i for the country-sector. Our dependent variable is real value added of country-sector i at time t, ln (𝑉𝐴𝑖𝑡), conventional inputs are capital and labour, measured as real capital stock (accumulated gross fixed capital formation), ln(𝐾𝑖𝑡), and total hours worked, ln (𝐿𝑖𝑡). Our primary variable of interest is the real R&D stock, ln(𝑅𝐷𝑖𝑡) – as can be seen, all these variables are in logarithms. In the first part of the question, (a) to (c), we focus on the returns to R&D, i.e. the statistical significance and magnitude of the R&D stock coefficient. Only part (d) uses tools from macro panel econometrics.

a. Discuss the general advantages and potential disadvantages of panel data over single cross-section data. [9 marks]

b. Write down an empirical model for ln (𝑉𝐴𝑖𝑡) with the above regressors (labour, capital, R&D) paying close attention to the subscripts and notation in general: (i) for a pooled OLS model with year fixed effects, and (ii) for a country-sector fixed effects model with additional year fixed effects. Explain any specific features of these models in words. [8 marks]

c. Interpret the results in Table 18.1 columns (1) to (4) with regards to (i) the sum of the three technology parameters, (ii) the returns to R&D investment, and (iii) common suggestion that the capital coefficient should be ‘around 0.3’ – see Table notes for pointers. Economic and statistical interpretation! [8 marks]

d. The models in columns (5) and (6) of Table 18.1 present average estimates from heterogeneous panel estimators: the Pesaran and Smith (1995) MG and the Pesaran (2006) CCEMG. Discuss which assumptions made in regression models (1) to (4) are relaxed in these empirical implementations. Which of models (5) and (6) can flexibly capture spillovers or common shocks? Which of the six models is your preferred model and why? What do the empirical results for this preferred model imply for the returns to R&D investment? [10 marks]

 

https://apaxresearchers.com/storage/files/2024/05/22/10751-aL3_13_49_53_econ-2.pdf