## ECON-M 514 ECONOMETRICS II (4 CR.)

This course emphasizes using econo-metric techniques to analyze economic problems, using real data. The course will focus on multiple regression methods for analyzing data in economics and related disciplines. Extensions include regression with discrete random variables, instrumental variables regression, analysis of random experiments, and quasi-experiments, and regression with time series.

1 classes found

### Spring 2021

Regular Academic Session / Hybrid-Distance Video & Online

LEC 29960: Total Seats: 30 / Available: 26 / Waitlisted: 0

## Stationary Time Series

The stationary $\operatorname{ARMA}(p, q)$ can be written
$$\phi(L) y_{t}=\alpha+\theta(L) \epsilon_{t}$$
with $\phi(z)=1-\phi_{1} z-\phi_{2} z^{2}-\ldots-\phi_{p} z^{p}$, a polynomial whose roots lie outside the unit circle and $\theta(z)=1+\theta_{1} z+\theta_{2} z^{2}+\ldots+\theta_{q} z^{q}$ and $\epsilon_{t}$ a white noise innovation process with 0 mean and variance $\sigma^{2}$
These are very good at capturing patterns of serial correlation present in time series with relatively few parameters.
We have looked at identification, estimation, diagnostic testing and forecasting with these models.

A time series is (covariance or second-order) stationary if its expectation (or mean) $\mu$, variance $E\left(y_{t}-\mu\right)^{2} \equiv \gamma_{0} ;$ and, autocovariances $E\left[\left(y_{t}-\mu\right)\left(y_{t-k}-\mu\right)\right] \equiv \gamma_{k}$, $k=0, \pm 1, \pm 2, \ldots$, are finite and do not depend on $t$. In reality, however, plenty of economic time series are simply not-stationary – remember we had to difference industrial production before working with it.
All our estimation and testing (so far) depends on the series being stationary.
We will look at:

• Trends – Deterministic and Stochastic
• Testing for unit roots
• Seasonal patterns

## Deterministic Trends in Time Series

The simplest deterministic trend is a linear time trend model
$$y_{t}=\alpha+\beta t+\epsilon_{t}, t=1, \ldots, n$$

• When $\beta \neq 0$ this is non-stationary because $E\left(y_{t}\right)=\alpha+\beta t$ varies over time.
• It can be combined with a stationary $\operatorname{ARMA}(p, q)$
$$\phi(L) y_{t}=\alpha+\beta t+\theta(L) \epsilon_{t} . \quad \text { (7.19) }$$
These processes are trend stationary
• In (7.19) we have a regressor $x_{t}=t$. Ass $n \rightarrow \infty$,
$$\frac{1}{n} \sum_{t=1}^{n} x_{t}^{2}=\frac{1}{n} \sum_{t=1}^{n} t^{2}=\frac{n(n+1)(2 n+1)}{6 n} \rightarrow \infty$$
an OLS estimate of $\beta$ is consistent at a faster rate than $\sqrt{n}$.
• Often the deterministic trend is fitted by OLS first, then a stationary ARMA fitted to the residuals.

The simplest stochastic trend model is a random walk
$$y_{t}=y_{t-1}+\epsilon_{t}(7.20)$$

• The direction of the trend cannot predicted, $E\left(\Delta y_{t} \mid Y_{t-1}\right)=0$
• Note that $(7.20)$ can be written
$$y_{t}=y_{1}+\sum_{s=2}^{t} \epsilon_{s}$$
• It is not mean-reverting – shocks do not decay like in stationary ARMA.
• It follows that $\operatorname{var}\left(y_{t}\right)=(t-1) \sigma^{2}$, which depends on $t$.
• These processes are known as:
• difference stationary – $\Delta y_{t}=y_{t}-y_{t-1}$ is stationary;
• unit root Like an $A R(1)$ with a root on the unit circle; or
• integrated of order 1 .

# ARIMA models

Stochastic trends can be incorperated into our ARMA framework in an $\operatorname{ARIMA}(p, d, q)$ model
$$\phi(L)(1-L)^{d} y_{t}=\theta(L) \epsilon_{t},$$
where $d$, the order of integration, is an integer.

• Typically we test to find the order of $d$ – We will look at tests for determining $d$ in the next lecture.
• Then we will difference the data $d$ times until we are satisfied that it is stationary.
• Then we use methods from lecture 8 estimate $\phi(z)$ and $\theta(z)$
• Forecasting is then similar to forecasting stationary ARMA models.

Very often time series show distinct seasonal patterns that appear to repeat themselves over and over throughout the data.
There could be lots of reasons behind these:

• Typical ‘seasonal’ effects from climate and, e.g. harvest, summer holidays.
• The effects of annual customs Christmas, New Year, Easter.
• Effects of administrative procedures like tax or budgeting years.

Deterministic seasonal patterns can be captured by including seasonal dummies in, for example for quarterly data
$$y_{t}=\alpha+\beta t+\delta_{2} D_{2}+\delta_{3} D_{3}+\delta_{4} D_{4}+\epsilon_{t}$$
where $D_{i}$ takes value 1 in the $i$ ‘th quarter and 0 otherwise. Stochastic seasonal patterns can be captured by including seasonal lags in an ARMA or ARIMA specification, such as the $\operatorname{SARIMA}(p, d, q)$ model for quarterly data
$$\phi\left(L^{4}\right)\left(1-L^{4}\right)^{d} y_{t}=\theta\left(L^{4}\right) \epsilon_{t}$$
and the ‘airline model’
$$(1-L)\left(1-L^{4}\right) y_{t}=\left(1+\theta_{1} L\right)\left(1+\theta_{4} L^{4}\right) \epsilon_{t}$$
There are also statistical methods for seasonal adjustment such as a CENSUS X12 ARIMA.

Non-stationary models
Stochastic and Deterministic Trends
Seasonal models.

# Section B

Answer ALL THREE questions in section B. To get good marks your answer should contain clear statements that explain the logical structure of your argument.

1. Consider the time series process
$$y_{t}=\alpha+\phi y_{t-1}+\varepsilon_{t}$$
where $|\phi|<1$ and $\varepsilon_{t}$ follows an $\operatorname{ARCH}(1)$ process with $\varepsilon_{t} \mid Y_{t-1} \sim N\left(0, \sigma_{t}^{2}\right)$, where $Y_{t-1}=\left{y_{t-1}, y_{t-2}, \ldots\right}$ denotes the past history of the process and $$\sigma_{t}^{2}=\alpha_{0}+\alpha_{1} \varepsilon_{t-1}^{2}$$ with $\alpha_{0}>0$ and $\alpha_{1} \geq 0$
(a) Using the fact that we may write $\varepsilon_{t}=\eta_{t} \sigma_{t}$, where $\eta_{t} \sim N(0,1)$ is a sequence of independent standard normal variables, show that $\varepsilon_{t}$ is a white noise process with mean zero and (unconditional) variance $\alpha_{0} /\left(1-\alpha_{1}\right)$ as long as $\varepsilon_{t}$ is stationary. (7 marks)
(b) Show that $\varepsilon_{t}^{2}$ follows an $\mathrm{AR}(1)$ process. Hence or otherwise discuss the conditions under which $\varepsilon_{t}$ is stationary. (7 marks)
(c) Suppose the parameters $\alpha, \phi, \alpha_{0}, \alpha_{1}$ are known. Write down the one-step ahead forecast, $\hat{y}{n+1}=\mathrm{E}\left[y{n+1} \mid y_{n}, y_{n-1}, \ldots\right] .$ Derive the variance of the forecast error. ( 6 marks)
1. Consider the salary data of a sample of 258 male bank employees. The dependent variable MANAGERIAL takes the value 1 if worker $i$ is in a managerial role, and 0 otherwise. The regressors EDUC measures the number of completed years of education, MINORITY is a dummy taking the value 1 if the employee is a member of an ethnic minority and PREVEXP measures previous work experience, in months. A researcher employs a logit model, $P\left(y_{i}=1\right)=\Lambda\left(\mathbf{x}{i}^{\prime} \boldsymbol{\beta}\right)$, where $\mathrm{x}{i}$ contains the regressors for worker $i$ and $\Lambda(t)=\frac{1}{1+e^{-t}}$ the logistic function.
(a) As a first attempt, the researcher estimates the following model.
\begin{tabular}{l}
\hline \hline Table 9.1. Dependent Variable: MANAGERIAL \
Method: ML – Binary Logit & \
\hline Variable & Coefficient & std error & Prob. \
C & $-26.95253$ & $4.400895$ & $0.0000$ \
EDUC & $1.674803$ & $0.280049$ & $0.0000$ \
MINORITY & $-2.395242$ & $0.847981$ & $0.0047$ \
PREVEXP & $0.003865$ & $0.003078$ & $0.2092$ \
\hline Mean dependent var & $0.286822$ & Akaike info criterion & $0.522420$ \
Log likelihood & $-63.39219$ & Avg. log likelihood & $-0.245706$ \
\hline
\end{tabular}
Comment briefly on the signs and significance of the coefficients in this model. Are they in line with your expectations? (4 marks)
(b) The researcher re-estimates the model, omitting the variable PREVEXP.
\begin{tabular}{l}
$\overline{\hline \text { Table 9.2. Dependent Variable: MANAGERIAL }}$ \
Method: ML – Binary Logit & \
\hline Variable & Coefficient & std error & Prob. \
C & $-26.21472$ & $4.311652$ & $0.0000$ \
EDUC & $1.644798$ & $0.276714$ & $0.0000$ \
MINORITY & $-2.119683$ & $0.793999$ & $0.0076$ \
\hline Mean dependent var & $0.286822$ & Akaike info criterion & $0.520543$ \
Log likelihood & $-64.15011$ & Avg. log likelihood & $-0.248644$ \
\hline
\end{tabular}
Using all relevant information from the tables, discuss which of the two models you prefer. Use the second model to estimate the probability that a male employee not from an ethinic minority with 15 years of education is in a managerial position. What might explain the increase in the estimated coefficient MINORITY when the variable PREVEXP is excluded from the regression?
(10 marks)
(c) According to the regression in part $9 b$, the probability that a male employee has a management job depends on his time in education. Explain carefully why that probability does not increase by $1.645$ for every additional year of education. Calculate an estimate of the marginal effect of an extra year of education for an ethnic minority male with 17 years of education. (6 marks)