Time Series Data

• Time series data: Data collected for a single entity at multiple points in time.
• This data type can be used to answer quantitative questions for which cross-sectional data are inadequate.
• Dynamic causal effect of variable \(x\) on dependent variable \(y\) over time.
• Example: What is the effect of a law requiring passengers to wear seatbelts on traffic fatalities?
• Forecasting the value of a variable at a later date.
• Example: What will next month’s inflation rate be?
• Using time series data allows us to extend the questions we can now answer. However, it poses special challenges, and overcoming them requires new techniques.

Example: Baby’s Names

Example: Unemployment in Austria

Example: GDP in Austria

Time Series

• Time series data are observations indexed according to time.
• It implies that the index has an ordering. Observations are not exchangeable.
• The observation on the time series variable \(y\) made at date \(t\) is denoted \(y_t\).
• The total number of observations is denoted \(T\).
• The interval (of time) between observation \(t\) and \(t+1\) is some unit of weeks, months, quarters, or even years.

Lags

• Special terminology and notation are used to indicate future and past values of \(y\).
• The value of \(y\) in the previous period is called its first lagged value or first lag, and is denoted \(y_{t-1}\).
• Its j-th lagged value is its value \(j\) periods ago, denoted \(y_{t-j}\).
• Similarly, \(y_{t+1}\) denotes the value of \(y\) one period into the future.

First Differences

• The change of value of \(y\) between period \(t-1\) and \(t\) is \(y_{t}-y_{t-1}\)
• We call it the first difference in the variable \(y_t\)
• We will use the following notation: \(\Delta y = y_t - y_{t-1}\)

Detrending: Fixed and Random Variation

Time series methods focus on modelling random components. But, there may also be fixed variation: seasonality of GDP, or weekday effects.

We are looking to remove fixed components (such as a trend or seasonal patterns) and focus on random components of a time series. There are many ways to achieve this:

• growth rates (e.g. year-to-year, quarter-to-quarter)
• linear trends (e.g. linear growth, demeaning seasons)
• filters (e.g. bandpass filter, Hodrick-Prescott filter)

For example, many economic time series are analyzed after computing their (change in) logarithms, as they exhibit exponential growth.

Detrending: Examples

Autocorrelation in time series regressions

Consider the linear model and related assumptions:

\[ y_t = {x}_t'{\beta} + u_t, \qquad u_t \sim \mathcal{N}(0,\sigma^2). \]

1. \(\mathrm{E}(u)=0\),
2. \(\mathrm{Var}(u)=\sigma^2\), and
3. \(\mathrm{Cov}(u_t,u_s)=0 \ \text{for all } t \neq s\).

When \(y\) and \(x\) are time series, Assumption (3) is often violated. We need a sensible alternative.

Autocorrelation

• In time series, the value of \(y\) in one period is typically correlated with its value in the next period.
• The correlation of a series with its own lagged values is called autocorrelation.
• The first autocorrelation is the correlation between \(y_t\) and \(y_{t-1}\)
• The j-th autocorrelation is the correlation between \(y_t\) and \(y_{t-j}\)
• The j-th autocovariance is the covariance between \(y_t\) and \(y_{t-j}\)

\[ \text{$j^{\text{th}}$ autocovariance} \;=\; \operatorname{cov}(y_t, y_{t-j}) \]

\[ \text{$j^{\text{th}}$ autocorrelation} \;=\; \rho_j \;=\; \;=\; \frac{\operatorname{cov}(y_t, y_{t-j})}{\sqrt{\operatorname{var}(y_t)\operatorname{var}(y_{t-j})}} \]

Autocorrelation function

The autocorrelation function (ACF) is a useful summary for a stationary time series. It is defined as:

\[ \mathrm{ACF}(j) \;=\; \rho_j \;=\; \operatorname{corr}(y_t, y_{t-j}) \;=\; \frac{\operatorname{cov}(y_t, y_{t-j})}{\operatorname{var}(y_t)}. \]

The ACF is a common diagnostic tool that reveals interesting patterns of autocorrelation, but it may also indicate a lack of stationarity.

Autocorrelation of GDP

Durbin-Watson Statistic

• We can test for autocorrelation using the Durbin-Watson statistic: \[ d = \frac{\sum_{t=2}^{T}\left(\hat{u}_t - \hat{u}_{t-1}\right)^2}{\sum_{t=1}^{T}\hat{u}_t^{\,2}}. \]
• A value of \(d = 2\) indicates no autocorrelation.
• The statistic is bounded as \(d \in [0,4]\)
• Values smaller than two may indicate positive autocorrelation.
• Values larger than two may indicate negative autocorrelation.

Stationarity

• Time series forecasts use data on the past to forecast the future.
• Doing so presumes that the future is similar to the past (in the sense that the correlations)
• More generally, the distributions of the data in the future will be like they were in the past.
• In the context of regression with time series data, the idea that historical relationships can be generalized to the future is formalized by the concept of stationarity.
• Definition of stationarity: The probability distribution of the time series variable does not change over time.
• Under the assumption of stationarity, regression models estimated using past data can be used to forecast future values
• Stationarity can fail to hold for multiple reasons, in which case the time series is said to be nonstationary

The First-Order Autoregressive Model

• An autoregression expresses the conditional mean of a time series variable \(y_t\) as a linear function of its own lagged values.
• A first-order autoregression uses only one lag of \(y\) in this conditional expectation.

\[ \mathbb{E}\left(y_t \mid y_{t-1}, y_{t-2}, \ldots \right) = \alpha_0 + \alpha_1 y_{t-1}. \]

• The first-order autoregression (AR(1)) model can be written in the familiar form of a regression model as

\[ y_t = \alpha_0 + \alpha_1 y_{t-1} + \varepsilon_t, \]

The First-Order Autoregressive Model (2)

• The unknown population coefficients \(\alpha_0\) and \(\alpha_1\) can be estimated by ordinary least square (OLS).
• How to estimate \(\alpha_0\) and \(\alpha_1\) might initially seem puzzling: Unlike a cross-sectional regression with \(x\) on the right-hand side, we have \(y\) on both sides.
• The solution to this puzzle is to realize that the variable \(y_{t-1}\) on the right-hand side differs from the dependent variable \(y_t\) because the regressor is the first lag of \(y\).
• That is, \(x\) is the first lag of \(y\).

Forecasts and forecast errors

• If the population coefficients were known, then the one-step-ahead forecast of \(y_{t+1}\), made using data through date \(t\), would be: \[ y_{t+1\mid t} = \alpha_0 + \alpha_1 y_t \]
• Although \(\alpha_0\) and \(\alpha_1\) are unknown, we can use their OLS estimates instead. The forecast based on the AR(1) model is: \[ \widehat{y}_{t+1\mid t} = \widehat{\alpha}_0 + \widehat{\alpha}_1 y_t, \]
• \(\widehat{\alpha}_0\) and \(\widehat{\alpha}_1\) are estimated using historical data through time \(t\).
• The forecast error is: \(y_{t+1} - \widehat{y}_{t+1\mid t}\)

The \(p\)-th-Order Autoregressive Model (1)

• The AR(1) model uses \(y_{t-1}\) to forecast \(y_t\), but doing so ignores potentially useful information in the more distant past.
• One way to incorporate this information is to include additional lags in the AR(1) model; this yields the \(p\)th-order autoregressive model.
• The model represents \(y_t\) as a linear function of \(p\) of its lagged values.
• In the AR(\(p\)) model, the regressors are \(y_{t-1}, y_{t-2}, \ldots, y_{t-p}\), and intercept.
• The number of lags, \(p\), included in an In the AR(\(p\)) model is called the order (or lag length) of the autoregression.

The \(p\)-th-Order Autoregressive Model (2)

The \(p\)-th-order autoregressive (AR(\(p\))) model represents the conditional expectation of \(y_t\) as a linear function of \(p\) of its lagged values:

\[ y_t = \alpha_0 + \alpha_1 y_{t-1} + \alpha_2 y_{t-2} + \cdots + \alpha_p y_{t-p} + \varepsilon_t. \]

If \(y_t\) follows an AR(\(p\)), then the oracle one-step-ahead forecast of \(y_{t+1}\) based on \(y_t, y_{t-1}, \ldots\) is

\[ y_{t+1\mid t} = \alpha_0 + \alpha_1 y_t + \alpha_2 y_{t-1} + \cdots + \alpha_p y_{t-p+1}. \]

The Autoregressive Distributed Lag (ADL) Model

An autoregressive distributed lag (ADL) model is:

• Autoregressive because lagged values of the dependent variable \(y_t\) are included as regressors.
• Distributed lag because the regression also includes multiple lags of an additional predictor \(x_t\).

An ADL model with \(p\) lags of \(y_t\) and \(q\) lags of \(x_t\), denoted ADL(\(p,q\)), is

\[ y_t = \alpha_0 + \alpha_1 y_{t-1} + \alpha_2 y_{t-2} + \cdots + \alpha_p y_{t-p} + \delta_1 x_{t-1} + \delta_2 x_{t-2} + \cdots + \delta_q x_{t-q} + \varepsilon_t. \]

Here, \(\alpha_0,\alpha_1,\ldots,\alpha_p,\delta_1,\ldots,\delta_q\) are unknown coefficients and \(\varepsilon_t\) is the error term, with

\[ \mathrm{E}\left(\varepsilon_t \mid y_{t-1},y_{t-2},\ldots,x_{t-1},x_{t-2},\ldots \right)=0. \]

General Time Series Regression with Additional Predictors

The general time series regression model allows for \(k\) additional predictors, where \(q_1\) lags of the first predictor are included, \(q_2\) lags of the second predictor are included, and so forth:

\[ \begin{aligned} y_t &= \alpha_0 + \alpha_1 y_{t-1} + \alpha_2 y_{t-2} + \cdots + \alpha_p y_{t-p} \\ &\quad + \delta_{11} x_{1,t-1} + \delta_{12} x_{1,t-2} + \cdots + \delta_{1q_1} x_{1,t-q_1} \\ &\quad + \cdots \\ &\quad + \delta_{k1} x_{k,t-1} + \delta_{k2} x_{k,t-2} + \cdots + \delta_{kq_k} x_{k,t-q_k} + \varepsilon_t . \end{aligned} \]

Assumptions

1. \(\mathrm{E}\left(\varepsilon_t \mid y_{t-1},y_{t-2},\ldots, x_{1,t-1},x_{1,t-2},\ldots, x_{k,t-1},x_{k,t-2},\ldots\right)=0\);
2. 1. The random variables \((y_t, x_{1t}, \ldots, x_{kt})\) have a stationary distribution, and
      
   2. \((y_t, x_{1t}, \ldots, x_{kt})\) and \((y_{t-j}, x_{1,t-j}, \ldots, x_{k,t-j})\) become independent as \(j\) gets large;
3. Large outliers are unlikely: \(y_t, x_{1t}, \ldots, x_{kt}\) have nonzero, finite fourth moments; and
4. There is no perfect multicollinearity.

Under the assumptions 1-4, inference on the regression coefficients using OLS proceeds in the same way as it usually does for cross-sectional data.

Assumption 1, 3, 4

• The first assumption is that \(\varepsilon_t\) has conditional mean zero given the history of all the regressors: \[ \mathrm{E}\left(\varepsilon_t \mid y_{t-1},y_{t-2},\ldots, x_{1,t-1},x_{1,t-2},\ldots, x_{k,t-1},x_{k,t-2},\ldots\right)=0. \]
• This assumption extends the assumption used in the AR and ADL models and implies that the forecast of \(y_t\), using all past values of \(y\) and the \(x\)’s, is given by the regression.
• The third assumption no outliers and fourth assumption no perfect multicollinearity are the same as for cross-sectional data.

Assumption 2

For time series regression, the i.i.d. assumption is replaced by a more appropriate assumption with two parts:

(a) Stationarity: The data are drawn from a stationary distribution, so the distribution of the time series today is the same as its distribution in the past. The joint distribution of the variables (including lags) does not change over time.

(b) Weak dependence: The random variables become approximately independent when they are separated by long periods of time. Weak dependence ensures that, in large samples, there is enough randomness for the law of large numbers and the central limit theorem to hold.

Determining the Order of an Autoregression

• Choosing the order p of an autoregression requires balancing the marginal benefit of including more lags against the marginal cost of additional estimation error.
• If the order of an estimated autoregression is too low, you will omit potentially valuable information contained in the more distant lagged values.
• If it is too high, you will be estimating more coefficients than necessary, which in turn introduces additional estimation error into your forecasts.

The F-statistic approach

• One approach to choosing \(p\) is to start with a model with many lags and to perform hypothesis tests on the final lag.
• For example, you might start by estimating an AR(6) and test whether the coefficient on the sixth lag is significant at the 5% level; if not, drop it and estimate an AR(5), test the coefficient on the fifth lag, and so forth.
• The drawback to this method is that it will tend to produce large models.
• Even if the true AR order is five, so the sixth coefficient is 0, a 5% test using the t-statistic will incorrectly reject this null hypothesis 5% of the time just by chance. Thus, if the true value of \(p\) is five, this method will estimate \(p\) to be six 5% of the time.

The Bayes Information Criterion (BIC) (1)

• One way to estimate \(p\) is by minimizing an information criterion, e.g., the BIC, which is:

\[ \text{BIC}(p) = \ln \left( \frac{SSR(p)}{T} \right) + (p + 1) \frac{\ln(T)}{T}, \]

• \(SSR(p)\) is the sum of squared residuals of the estimated AR(\(p\)).
• The BIC estimator of \(p\), \(\hat{p}\), is the value that minimizes \(\text{BIC}(p)\) among the possible choices \(p = 0, 1, \ldots, p_{\max}\)
• \(p_{\max}\) is the largest value of \(p\) considered and \(p = 0\) corresponds to the model that contains only an intercept.

The Bayes Information Criterion (BIC) (2)

• Because the regression coefficients are estimated by OLS, the SSR necessarily decreases (or at least does not increase) when you add a lag.
• In contrast, the second term is the number of estimated regression coefficients (the number of lags, \(p\), plus one for the intercept) times the factor \(\frac{\ln(T)}{T}\).
• This second term increases when you add a lag and thus provides a penalty for including another lag.
• The BIC trades off these two forces so that the number of lags that minimizes the BIC is a consistent estimator of the true lag length.
• The BIC helps decide precisely how large the increase in the \(R^2\) must be to justify including the additional lag.

The Akaike Information Criterion (AIC)

• Another information criterion is the Akaike information criterion (AIC): \[ \text{AIC}(p) = \ln \left( \frac{SSR(p)}{T} \right) + \frac{(p + 1)^2}{T}, \]
• The difference between the AIC and the BIC is that the term \(\mathrm{ln}(T)\) in the BIC is replaced by \(2\) in the AIC, so the second term in the AIC is smaller.
• The second term in the AIC is not large enough to ensure that the correct lag length is chosen, even in large samples, so the AIC estimator of \(p\) is not consistent.
• If you are concerned that the BIC might yield a model with too few lags, the AIC provides a reasonable alternative

Lag Length Selection with Multiple Predictors

The trade-off involved with lag length choice in the general time series regression model with multiple predictors is similar to that in an autoregression:

• Using too few lags can decrease forecast accuracy because valuable information is lost
• Adding lags increases estimation error.

The choice of lags must balance the benefit of using additional information against the cost of estimating the additional coefficients.

The F-Statistic Approach

• As in the univariate autoregression, one way to determine the number of lags is to use F-statistics to test joint hypotheses that sets of coefficients are equal to 0.
• If the number of models being compared is small, then this F-statistic method is easy to use.
• In general, however, the F-statistic method can produce models that are large and thus have considerable estimation error

Information Criteria for Model Selection

As in an autoregression, the BIC and the AIC can be used to estimate the number of lags and variables in a time series regression model with multiple predictors. If the regression model has \(K\) coefficients (including the intercept), the BIC is

\[ \text{BIC}(K) = \ln\!\left(\frac{SSR(K)}{T}\right) + K\frac{\ln(T)}{T}. \]

The AIC is defined in the same way, but with \(2\) replacing \(\ln(T)\) in the second term:

\[ \text{AIC}(K) = \ln\!\left(\frac{SSR(K)}{T}\right) + K\frac{2}{T}. \]

For each candidate model, the BIC (or the AIC) can be evaluated, and the model with the lowest value of the BIC (or the AIC) is the preferred model, based on the information criterion.

Nonstationarity

• Until now we have assumed that the dependent variable and the regressors are stationary.
• If this is not the case—that is, if the dependent variable and/or the regressors are nonstationarity—then conventional hypothesis tests, confidence intervals, and forecasts can be unreliable.
• The precise problem created by nonstationarity, and the solution to that problem, depends on the nature of that nonstationarity
• There are two types of nonstationarity frequently encountered in economic time series: trends and breaks.

What Is a Trend?

• A trend is a persistent long-term movement of a variable over time. A time series variable fluctuates around its trend
• There are two types of trends in time series data: deterministic and stochastic.
• A deterministic trend is a nonrandom function of time.
• A stochastic trend is random and varies over time.
• It is more appropriate to model economic time series as having stochastic rather than deterministic trends.

The Random Walk Model of a Trend

• The basic idea of a random walk is that the value of the series tomorrow is its value today plus an unpredictable change.
• Because the path followed by \(y_t\) consists of random “steps” \(\varepsilon_t\), that path is a random walk.
• The conditional mean of \(y_t\) based on data through time \(t-1\) is \(y_{t-1}\). That is, because \[ \mathrm{E}\left(\varepsilon_t \mid y_{t-1}, y_{t-2}, \ldots \right)=0, \] it follows that \[ \mathrm{E}\left(y_t \mid y_{t-1}, y_{t-2}, \ldots \right)=y_{t-1}. \]
• if \(y_t\) follows a random walk, then the best forecast of tomorrow’s value is its value today.
• If \(y_t\) follows a random walk, its variance increases over time. Because it does not have a constant variance, a random walk is nonstationary

Random Walk with Drift

• Some series, such as the logarithm of U.S. GDP, have an obvious upward tendency. In that case, the best forecast must include an adjustment for the tendency of the series to increase.
• This leads to an extension of the random walk model that includes a tendency to move, or drift, in one direction or the other. This extension is referred to as a random walk with drift: \[ y_t = \alpha_0 + y_{t-1} + \varepsilon_t, \] \[ \mathrm{E}\left(\varepsilon_t \mid y_{t-1}, y_{t-2}, \ldots \right)=0, \]
• \(\alpha_0\) is the drift in the random walk.
• If \(\alpha_0>0\), then \(y_t\) increases on average.

Unit Root (1)

• The random walk model is a special case of the AR(1) model in which \(\alpha_1 = 1\).
• If \(y_t\) follows an AR(1) with \(\alpha_1 = 1\), then \(y_t\) contains a stochastic trend and is nonstationary.
• If \(|\alpha_1| < 1\) and \(\varepsilon_t\) is stationary, then the joint distribution of \(y_t\) and its lags does not depend on \(t\), \(y_t\) is stationary.
• For an AR(\(p\)) to be stationary is more complicated. Its formal statement involves the roots of the polynomial \[ 1 - \alpha_1 z - \alpha_2 z^2 - \alpha_3 z^3 - \cdots - \alpha_p z^p. \]
• For an AR(\(p\)) to be stationary, all roots of this polynomial must be greater than 1 in absolute value.

Unit Root (2)

• In the special case of an AR(1), the root solves \[ 1 - \alpha_1 z = 0, \]
• so the root is \[ z = \frac{1}{\alpha_1}. \]
• Saying that the root must be greater than 1 in absolute value is equivalent to \(|\alpha_1| < 1\).
• If an AR(\(p\)) has a root equal to 1, the series is said to have a unit autoregressive root (or a unit root)
• If \(y_t\) has a unit root, then it contains a stochastic trend.

Problems Caused by Stochastic Trends

If a regressor has a stochastic trend (has a unit root), then OLS inference can be misleading.

• the usual OLS \(t\)-statistic can have a nonnormal distribution, even in large samples
• the estimate of the autoregressive coefficient is biased toward 0
• spurious regression: two independent series with stochastic trends can appear to be related

As a result, conventional confidence intervals and hypothesis tests are not valid in the usual way.

Detecting Stochastic Trends: The Dickey–Fuller Test

• The hypothesis of a stochastic trend can be tested using a Dickey–Fuller test. For the AR(1) model: \[ y_t = \alpha_0 + \alpha_1 y_{t-1} + \varepsilon_t, \]
• the null hypothesis that \(y_t\) has a stochastic trend is \[ H_0:\alpha_1=1 \qquad \text{vs.} \qquad H_1:\alpha_1<1. \]
• Let \(\rho = \alpha_1 - 1\). Then the model can be rewritten as \[ \Delta y_t = \alpha_0 + \rho y_{t-1} + \varepsilon_t, \]
• The hypotheses become: \(H_0:\rho=0 \qquad \text{vs.} \qquad H_1:\rho<0.\)
• The OLS \(t\)-statistic for testing \(\rho=0\) is called the Dickey–Fuller statistic.

Critical values for the ADF Statistic

• Under the null hypothesis of a unit root, the ADF statistic does not have a normal distribution, even in large samples, so standard critical values cannot be used.
• Because the alternative hypothesis is \(\rho < 0\), the ADF test is one-sided.
• Studies of the ADF statistic suggest that it is better to have too many lags than too few
• it is then recommended to use the AIC instead of the BIC to estimate p for the ADF statistic
• The most reliable way to handle a trend in a series is to transform the series so that it does not have the trend.
• If the series has a stochastic trend, then its difference does not

Nonstationarity II: Breaks

• A second type of nonstationarity arises when the population regression function changes over the sample period.
• These changes, or breaks, can make inference and forecasting misleading if they are ignored.
• Breaks can occur because of changes in policy, economic structure, or industry conditions.
• A break can arise from:
• A discrete change in the regression coefficients at a specific date;
• A gradual change in the coefficients over time.
• One way to detect breaks is to test for discrete changes, or breaks, in the regression coefficients.
• How this is done depends on whether the break date is known

Testing for a Break at a Known Date (1)

• If the date of the hypothesized break is known, the null hypothesis of no break can be tested using a binary variable interaction regression.
• For simplicity, consider an ADL(1, 1) model with an intercept, a single lag of \(y_t\), and a single lag of \(x_t\).
• Let \(\tau\) denote the break date, and let \(D_t(\tau)\) be a binary variable equal to 0 before the break and 1 after.
• The regression including the binary break indicator and all interaction terms is:

\[ y_t = \alpha_0 + \alpha_1 y_{t-1} + \delta_1 x_{t-1} + \gamma_0 D_t(\tau) + \gamma_1 [D_t(\tau) \times y_{t-1}] + \gamma_2 [D_t(\tau) \times x_{t-1}] + \varepsilon_t \]

Testing for a Break at a Known Date (2)

• If there is no break, then the population regression function is the same over both parts of the sample, so the terms involving the break indicator \(D_t(\tau)\) do not enter.
• The null hypothesis of no break implies \(\gamma_0 = \gamma_1 = \gamma_2 = 0\).
• The alternative hypothesis is that there is a break, and the population regression function is different before and after the break date, implying at least one of the \(\gamma\)’s is nonzero.
• The hypothesis of a break can be tested using the F-statistic testing \(\gamma_0 = \gamma_1 = \gamma_2 = 0\) against the alternative.
• This is often referred to as the Chow test.

Testing for a Break at an Unknown Date

• Often, the date of a possible break is unknown or known only within a range.
• You might suspect that a break occurred between two dates, \(\tau_0\) and \(\tau_1\).
• The Chow test can be extended to test for breaks at all possible dates \(\tau\) between \(\tau_0\) and \(\tau_1\) and then use the largest resulting \(F\)-statistics to test for a break at an unknown date.
• This modified Chow test is called the Quandt likelihood ratio (QLR) statistic (sometimes the sup-Wald statistic).
• Because the QLR statistic is the largest of many \(F\)-statistics, its distribution is not the same as an individual \(F\)-statistic.
• Instead, the critical values for the QLR statistic must be obtained from a special distribution.
• The QLR test can detect a single discrete break, multiple discrete breaks, and/or slow evolution of the regression function.

Detecting Breaks Using Pseudo Out-of-Sample Forecasts

• The ultimate test of a forecasting model is its out-of-sample performance: its forecasting performance in “real time,” after the model has been estimated.
• Pseudo out-of-sample forecasting simulates the real-time performance of a forecasting model and can be used to detect breaks near the end of the sample.
• The most direct and often most useful way to do so is via a time series plot of the in-sample predicted values, the pseudo out-of-sample forecasts, and the actual values of the series.
• A visible deterioration of the forecasts in the pseudo out-of-sample period is a red flag warning of a possible breakdown of the forecasting model.

Example: Do you see a Trend and/or a Break?

Summary

• Regression models used for forecasting need not have a causal interpretation.
• A time series variable generally is correlated with one or more of its lagged values; that is, it is serially correlated.
• The accuracy of a forecast is measured by its mean squared forecast error.
• An autoregression of order \(p\) is a linear multiple regression model in which the regressors are the first \(p\) lags of the dependent variable.
• The coefficients of an AR(\(p\)) model can be estimated by OLS, and the estimated regression function can be used for forecasting.
• The lag order \(p\) can be estimated using an information criterion such as the BIC or the AIC.
• Adding other variables and their lags to an autoregression can improve forecasting performance.
• Under the least squares assumptions for prediction with time series regression, the OLS estimators have normal distributions in large samples, and statistical inference proceeds the same way as for cross-sectional data.

Forecasting Multiple Variables

One Model per Variable or One Model for All?

Why Do We Call It a “Vector” Autoregression?

\[ y_t = a_{10} + a_{11} y_{t-1} + a_{12} x_{t-1} + \varepsilon_{1t} \]

\[ x_t = a_{20} + a_{21} y_{t-1} + a_{22} x_{t-1} + \varepsilon_{2t} \]

Three Time Series

General Reduced-Form VAR(p)

Estimation and Hypothesis Tests

\[ \begin{aligned} \boldsymbol{y}_t &= \boldsymbol{a}_0 + \boldsymbol{A}_1\boldsymbol{y}_{t-1} + \dots + \boldsymbol{A}_p\boldsymbol{y}_{t-p} + \boldsymbol{\varepsilon}_t, \\ \boldsymbol{\varepsilon}_t &\sim \mathcal{N}_M(\boldsymbol{0},\boldsymbol{\Sigma}), \end{aligned} \]

This can be estimated using multiple techniques.

Estimating a VAR: Data

Estimating a VAR: Plotting

Estimating a VAR: Results

Causal Analysis

Granger Causality

Structural VARs

Cointegration

Defining Cointegration

Cointegration

Error Correction

Testing for Cointegration

Testing for Cointegration with a Known Coefficient

Testing for Cointegration with an Unknown Coefficient

Variance over Time

Volatility Clustering

Autoregressive Conditional Heteroskedasticity (ARCH)

Generalized ARCH (GARCH)

Time and Space

Tobler’s First Law of Geography

Which Ways Can Things be Related?

Spatial Weights

Contiguity

Distance

Back to Spatial Autocorrelation

Moran’s I

Space Is Everywhere

Spillovers

A Vicious Cycle

Outlook