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Principles of Econometrics, 4t h EditionPage 1 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Chapter 12 Regression with Time-Series Data: Nonstationary Variables Walter R. Paczkowski Rutgers University

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Principles of Econometrics, 4t h EditionPage 2 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.1 Stationary and Nonstationary Variables 12.2 Spurious Regressions 12.3 Unit Root Tests for Nonstationarity 12.4 Cointegration 12.5 Regression When There Is No Cointegration Chapter Contents

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Principles of Econometrics, 4t h EditionPage 3 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The aim is to describe how to estimate regression models involving nonstationary variables – The first step is to examine the time-series concepts of stationarity (and nonstationarity) and how we distinguish between them. – Cointegration is another important related concept that has a bearing on our choice of a regression model

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Principles of Econometrics, 4t h EditionPage 4 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.1 Stationary and Nonstationary Variables

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Principles of Econometrics, 4t h EditionPage 5 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The change in a variable is an important concept – The change in a variable y t, also known as its first difference, is given by Δy t = y t – y t-1. Δy t is the change in the value of the variable y from period t - 1 to period t 12.1 Stationary and Nonstationary Variables

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Principles of Econometrics, 4t h EditionPage 6 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.1 Stationary and Nonstationary Variables FIGURE 12.1 U.S. economic time series

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Principles of Econometrics, 4t h EditionPage 7 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.1 Stationary and Nonstationary Variables FIGURE 12.1 (Continued) U.S. economic time series

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Principles of Econometrics, 4t h EditionPage 8 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Formally, a time series y t is stationary if its mean and variance are constant over time, and if the covariance between two values from the series depends only on the length of time separating the two values, and not on the actual times at which the variables are observed 12.1 Stationary and Nonstationary Variables

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Principles of Econometrics, 4t h EditionPage 9 Chapter 12: Regression with Time-Series Data: Nonstationary Variables That is, the time series y t is stationary if for all values, and every time period, it is true that: 12.1 Stationary and Nonstationary Variables Eq. 12.1a Eq. 12.1b Eq. 12.1c

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Principles of Econometrics, 4t h EditionPage 10 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.1 Stationary and Nonstationary Variables Table 12.1 Sample Means of Time Series Shown in Figure 12.1

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Principles of Econometrics, 4t h EditionPage 11 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Nonstationary series with nonconstant means are often described as not having the property of mean reversion – Stationary series have the property of mean reversion 12.1 Stationary and Nonstationary Variables

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Principles of Econometrics, 4t h EditionPage 12 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The econometric model generating a time-series variable y t is called a stochastic or random process – A sample of observed y t values is called a particular realization of the stochastic process It is one of many possible paths that the stochastic process could have taken 12.1 Stationary and Nonstationary Variables 12.1.1 The First-Order Autoregressive Model

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Principles of Econometrics, 4t h EditionPage 13 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The autoregressive model of order one, the AR(1) model, is a useful univariate time series model for explaining the difference between stationary and nonstationary series: – The errors v t are independent, with zero mean and constant variance, and may be normally distributed – The errors are sometimes known as ‘‘shocks’’ or ‘‘innovations’’ 12.1 Stationary and Nonstationary Variables 12.1.1 The First-Order Autoregressive Model Eq. 12.2a

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Principles of Econometrics, 4t h EditionPage 14 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.1 Stationary and Nonstationary Variables 12.1.1 The First-Order Autoregressive Model FIGURE 12.2 Time-series models

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Principles of Econometrics, 4t h EditionPage 15 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.1 Stationary and Nonstationary Variables 12.1.1 The First-Order Autoregressive Model FIGURE 12.2 (Continued) Time-series models

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Principles of Econometrics, 4t h EditionPage 16 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The value ‘‘zero’’ is the constant mean of the series, and it can be determined by doing some algebra known as recursive substitution – Consider the value of y at time t = 1, then its value at time t = 2 and so on – These values are: 12.1 Stationary and Nonstationary Variables 12.1.1 The First-Order Autoregressive Model

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Principles of Econometrics, 4t h EditionPage 17 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The mean of y t is: Real-world data rarely have a zero mean – We can introduce a nonzero mean μ as: – Or 12.1 Stationary and Nonstationary Variables 12.1.1 The First-Order Autoregressive Model Eq. 12.2b

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Principles of Econometrics, 4t h EditionPage 18 Chapter 12: Regression with Time-Series Data: Nonstationary Variables With α = 1 and ρ = 0.7: 12.1 Stationary and Nonstationary Variables 12.1.1 The First-Order Autoregressive Model

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Principles of Econometrics, 4t h EditionPage 19 Chapter 12: Regression with Time-Series Data: Nonstationary Variables An extension to Eq. 12.2a is to consider an AR(1) model fluctuating around a linear trend: (μ + δt) – Let the ‘‘de-trended’’ series (y t -μ - δt) behave like an autoregressive model: Or: 12.1 Stationary and Nonstationary Variables 12.1.1 The First-Order Autoregressive Model Eq. 12.2c

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Principles of Econometrics, 4t h EditionPage 20 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Consider the special case of ρ = 1: – This model is known as the random walk model These time series are called random walks because they appear to wander slowly upward or downward with no real pattern the values of sample means calculated from subsamples of observations will be dependent on the sample period – This is a characteristic of nonstationary series 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models Eq. 12.3a

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Principles of Econometrics, 4t h EditionPage 21 Chapter 12: Regression with Time-Series Data: Nonstationary Variables We can understand the ‘‘wandering’’ by recursive substitution: 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models

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Principles of Econometrics, 4t h EditionPage 22 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The term is often called the stochastic trend – This term arises because a stochastic component v t is added for each time t, and because it causes the time series to trend in unpredictable directions 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models

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Principles of Econometrics, 4t h EditionPage 23 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Recognizing that the v t are independent, taking the expectation and the variance of y t yields, for a fixed initial value y 0 : – The random walk has a mean equal to its initial value and a variance that increases over time, eventually becoming infinite 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models

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Principles of Econometrics, 4t h EditionPage 24 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Another nonstationary model is obtained by adding a constant term: – This model is known as the random walk with drift 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models Eq. 12.3b

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Principles of Econometrics, 4t h EditionPage 25 Chapter 12: Regression with Time-Series Data: Nonstationary Variables A better understanding is obtained by applying recursive substitution: 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models

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Principles of Econometrics, 4t h EditionPage 26 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The term tα a deterministic trend component – It is called a deterministic trend because a fixed value α is added for each time t – The variable y wanders up and down as well as increases by a fixed amount at each time t 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models

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Principles of Econometrics, 4t h EditionPage 27 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The mean and variance of y t are: 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models

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Principles of Econometrics, 4t h EditionPage 28 Chapter 12: Regression with Time-Series Data: Nonstationary Variables We can extend the random walk model even further by adding a time trend: 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models Eq. 12.3c

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Principles of Econometrics, 4t h EditionPage 29 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The addition of a time-trend variable t strengthens the trend behavior: where we used: 12.1 Stationary and Nonstationary Variables 12.1.2 Random Walk Models

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Principles of Econometrics, 4t h EditionPage 30 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.2 Spurious Regressions

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Principles of Econometrics, 4t h EditionPage 31 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The main reason why it is important to know whether a time series is stationary or nonstationary before one embarks on a regression analysis is that there is a danger of obtaining apparently significant regression results from unrelated data when nonstationary series are used in regression analysis – Such regressions are said to be spurious 12.2 Spurious Regressions

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Principles of Econometrics, 4t h EditionPage 32 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Consider two independent random walks: – These series were generated independently and, in truth, have no relation to one another – Yet when plotted we see a positive relationship between them 12.2 Spurious Regressions

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Principles of Econometrics, 4t h EditionPage 33 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.2 Spurious Regressions FIGURE 12.3 Time series and scatter plot of two random walk variables

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Principles of Econometrics, 4t h EditionPage 34 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.2 Spurious Regressions FIGURE 12.3 (Continued) Time series and scatter plot of two random walk variables

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Principles of Econometrics, 4t h EditionPage 35 Chapter 12: Regression with Time-Series Data: Nonstationary Variables A simple regression of series one (rw 1 ) on series two (rw 2 ) yields: – These results are completely meaningless, or spurious The apparent significance of the relationship is false 12.2 Spurious Regressions

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Principles of Econometrics, 4t h EditionPage 36 Chapter 12: Regression with Time-Series Data: Nonstationary Variables When nonstationary time series are used in a regression model, the results may spuriously indicate a significant relationship when there is none – In these cases the least squares estimator and least squares predictor do not have their usual properties, and t-statistics are not reliable – Since many macroeconomic time series are nonstationary, it is particularly important to take care when estimating regressions with macroeconomic variables 12.2 Spurious Regressions

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Principles of Econometrics, 4t h EditionPage 37 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.3 Unit Root Tests for Stationarity

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Principles of Econometrics, 4t h EditionPage 38 Chapter 12: Regression with Time-Series Data: Nonstationary Variables There are many tests for determining whether a series is stationary or nonstationary – The most popular is the Dickey–Fuller test 12.3 Unit Root Tests for Stationarity

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Principles of Econometrics, 4t h EditionPage 39 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The AR(1) process y t = ρy t-1 + v t is stationary when |ρ| < 1 – But, when ρ = 1, it becomes the nonstationary random walk process – We want to test whether ρ is equal to one or significantly less than one Tests for this purpose are known as unit root tests for stationarity 12.3 Unit Root Tests for Stationarity 12.3.1 Dickey-Fuller Test 1 (No constant and No Trend)

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Principles of Econometrics, 4t h EditionPage 40 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Consider again the AR(1) model: – We can test for nonstationarity by testing the null hypothesis that ρ = 1 against the alternative that |ρ| < 1 Or simply ρ < 1 12.3 Unit Root Tests for Stationarity Eq. 12.4 12.3.1 Dickey-Fuller Test 1 (No constant and No Trend)

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Principles of Econometrics, 4t h EditionPage 41 Chapter 12: Regression with Time-Series Data: Nonstationary Variables A more convenient form is: – The hypotheses are: 12.3 Unit Root Tests for Stationarity Eq. 12.5a 12.3.1 Dickey-Fuller Test 1 (No constant and No Trend)

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Principles of Econometrics, 4t h EditionPage 42 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The second Dickey–Fuller test includes a constant term in the test equation: – The null and alternative hypotheses are the same as before 12.3 Unit Root Tests for Stationarity 12.3.2 Dickey-Fuller Test 2 (With Constant but No Trend) Eq. 12.5b

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Principles of Econometrics, 4t h EditionPage 43 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The third Dickey–Fuller test includes a constant and a trend in the test equation: – The null and alternative hypotheses are H 0 : γ = 0 and H 1 :γ < 0 (same as before) 12.3 Unit Root Tests for Stationarity 12.3.3 Dickey-Fuller Test 3 (With Constant and With Trend) Eq. 12.5c

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Principles of Econometrics, 4t h EditionPage 44 Chapter 12: Regression with Time-Series Data: Nonstationary Variables To test the hypothesis in all three cases, we simply estimate the test equation by least squares and examine the t-statistic for the hypothesis that γ = 0 – Unfortunately this t-statistic no longer has the t-distribution – Instead, we use the statistic often called a τ (tau) statistic 12.3 Unit Root Tests for Stationarity 12.3.4 The Dickey-Fuller Critical Values

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Principles of Econometrics, 4t h EditionPage 45 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.3 Unit Root Tests for Stationarity 12.3.4 The Dickey-Fuller Critical Values Table 12.2 Critical Values for the Dickey–Fuller Test

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Principles of Econometrics, 4t h EditionPage 46 Chapter 12: Regression with Time-Series Data: Nonstationary Variables To carry out a one-tail test of significance, if τ c is the critical value obtained from Table 12.2, we reject the null hypothesis of nonstationarity if τ ≤ τ c – If τ > τ c then we do not reject the null hypothesis that the series is nonstationary 12.3 Unit Root Tests for Stationarity 12.3.4 The Dickey-Fuller Critical Values

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Principles of Econometrics, 4t h EditionPage 47 Chapter 12: Regression with Time-Series Data: Nonstationary Variables An important extension of the Dickey–Fuller test allows for the possibility that the error term is autocorrelated – Consider the model: where 12.3 Unit Root Tests for Stationarity 12.3.4 The Dickey-Fuller Critical Values Eq. 12.6

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Principles of Econometrics, 4t h EditionPage 48 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The unit root tests based on Eq. 12.6 and its variants (intercept excluded or trend included) are referred to as augmented Dickey–Fuller tests – When γ = 0, in addition to saying that the series is nonstationary, we also say the series has a unit root – In practice, we always use the augmented Dickey–Fuller test 12.3 Unit Root Tests for Stationarity 12.3.4 The Dickey-Fuller Critical Values

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Principles of Econometrics, 4t h EditionPage 49 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.3 Unit Root Tests for Stationarity 12.3.5 The Dickey-Fuller Testing Procedures Table 12.3 AR processes and the Dickey-Fuller Tests

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Principles of Econometrics, 4t h EditionPage 50 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The Dickey-Fuller testing procedure: – First plot the time series of the variable and select a suitable Dickey-Fuller test based on a visual inspection of the plot If the series appears to be wandering or fluctuating around a sample average of zero, use test equation (12.5a) If the series appears to be wandering or fluctuating around a sample average which is nonzero, use test equation (12.5b) If the series appears to be wandering or fluctuating around a linear trend, use test equation (12.5c) 12.3 Unit Root Tests for Stationarity 12.3.5 The Dickey-Fuller Testing Procedures

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Principles of Econometrics, 4t h EditionPage 51 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The Dickey-Fuller testing procedure (Continued) : – Second, proceed with one of the unit root tests described in Sections 12.3.1 to 12.3.3 12.3 Unit Root Tests for Stationarity 12.3.5 The Dickey-Fuller Testing Procedures

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Principles of Econometrics, 4t h EditionPage 52 Chapter 12: Regression with Time-Series Data: Nonstationary Variables As an example, consider the two interest rate series: – The federal funds rate (F t ) – The three-year bond rate (B t ) Following procedures described in Sections 12.3 and 12.4, we find that the inclusion of one lagged difference term is sufficient to eliminate autocorrelation in the residuals in both cases 12.3 Unit Root Tests for Stationarity 12.3.6 The Dickey-Fuller Tests: An Example

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Principles of Econometrics, 4t h EditionPage 53 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The results from estimating the resulting equations are: – The 5% critical value for tau (τ c ) is -2.86 – Since -2.505 > -2.86, we do not reject the null hypothesis of non-stationarity. 12.3 Unit Root Tests for Stationarity 12.3.6 The Dickey-Fuller Tests: An Example

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Principles of Econometrics, 4t h EditionPage 54 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Recall that if y t follows a random walk, then γ = 0 and the first difference of y t becomes: – Series like y t, which can be made stationary by taking the first difference, are said to be integrated of order one, and denoted as I(1) Stationary series are said to be integrated of order zero, I(0) – In general, the order of integration of a series is the minimum number of times it must be differenced to make it stationary 12.3 Unit Root Tests for Stationarity 12.3.7 Order of Integration

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Principles of Econometrics, 4t h EditionPage 55 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The results of the Dickey–Fuller test for a random walk applied to the first differences are: 12.3 Unit Root Tests for Stationarity 12.3.7 Order of Integration

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Principles of Econometrics, 4t h EditionPage 56 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Based on the large negative value of the tau statistic (-5.487 < -1.94), we reject the null hypothesis that ΔF t is nonstationary and accept the alternative that it is stationary – We similarly conclude that ΔB t is stationary (-7.662 < -1.94) 12.3 Unit Root Tests for Stationarity 12.3.7 Order of Integration

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Principles of Econometrics, 4t h EditionPage 57 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.4 Cointegration

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Principles of Econometrics, 4t h EditionPage 58 Chapter 12: Regression with Time-Series Data: Nonstationary Variables As a general rule, nonstationary time-series variables should not be used in regression models to avoid the problem of spurious regression – There is an exception to this rule 12.4 Cointegration

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Principles of Econometrics, 4t h EditionPage 59 Chapter 12: Regression with Time-Series Data: Nonstationary Variables There is an important case when e t = y t - β 1 - β 2 x t is a stationary I(0) process – In this case y t and x t are said to be cointegrated Cointegration implies that y t and x t share similar stochastic trends, and, since the difference e t is stationary, they never diverge too far from each other 12.4 Cointegration

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Principles of Econometrics, 4t h EditionPage 60 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The test for stationarity of the residuals is based on the test equation: – The regression has no constant term because the mean of the regression residuals is zero. – We are basing this test upon estimated values of the residuals 12.4 Cointegration Eq. 12.7

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Principles of Econometrics, 4t h EditionPage 61 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.4 Cointegration Table 12.4 Critical Values for the Cointegration Test

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Principles of Econometrics, 4t h EditionPage 62 Chapter 12: Regression with Time-Series Data: Nonstationary Variables There are three sets of critical values – Which set we use depends on whether the residuals are derived from: 12.4 Cointegration Eq. 12.8a Eq. 12.8b Eq. 12.8c

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Principles of Econometrics, 4t h EditionPage 63 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Consider the estimated model: – The unit root test for stationarity in the estimated residuals is: 12.4 Cointegration Eq. 12.9 12.4.1 An Example of a Cointegration Test

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Principles of Econometrics, 4t h EditionPage 64 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The null and alternative hypotheses in the test for cointegration are: – Similar to the one-tail unit root tests, we reject the null hypothesis of no cointegration if τ ≤ τ c, and we do not reject the null hypothesis that the series are not cointegrated if τ > τ c 12.4 Cointegration 12.4.1 An Example of a Cointegration Test

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Principles of Econometrics, 4t h EditionPage 65 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Consider a general model that contains lags of y and x – Namely, the autoregressive distributed lag (ARDL) model, except the variables are nonstationary: 12.4 Cointegration 12.4.2 The Error Correction Model

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Principles of Econometrics, 4t h EditionPage 66 Chapter 12: Regression with Time-Series Data: Nonstationary Variables If y and x are cointegrated, it means that there is a long-run relationship between them – To derive this exact relationship, we set y t = y t-1 = y, x t = x t-1 = x and v t = 0 – Imposing this concept in the ARDL, we obtain: This can be rewritten in the form: 12.4 Cointegration 12.4.2 The Error Correction Model

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Principles of Econometrics, 4t h EditionPage 67 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Add the term -y t-1 to both sides of the equation: – Add the term – δ 0 x t-1 + δ 0 x t-1 : – Manipulating this we get: 12.4 Cointegration 12.4.2 The Error Correction Model

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Principles of Econometrics, 4t h EditionPage 68 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Or: – This is called an error correction equation – This is a very popular model because: It allows for an underlying or fundamental link between variables (the long-run relationship) It allows for short-run adjustments (i.e. changes) between variables, including adjustments to achieve the cointegrating relationship 12.4 Cointegration 12.4.2 The Error Correction Model Eq. 12.10

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Principles of Econometrics, 4t h EditionPage 69 Chapter 12: Regression with Time-Series Data: Nonstationary Variables For the bond and federal funds rates example, we have: – The estimated residuals are 12.4 Cointegration 12.4.2 The Error Correction Model

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Principles of Econometrics, 4t h EditionPage 70 Chapter 12: Regression with Time-Series Data: Nonstationary Variables The result from applying the ADF test for stationarity is: – Comparing the calculated value (-3.929) with the critical value, we reject the null hypothesis and conclude that (B, F) are cointegrated 12.4 Cointegration

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Principles of Econometrics, 4t h EditionPage 71 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.5 Regression with No-Cointegration

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Principles of Econometrics, 4t h EditionPage 72 Chapter 12: Regression with Time-Series Data: Nonstationary Variables How we convert nonstationary series to stationary series, and the kind of model we estimate, depend on whether the variables are difference stationary or trend stationary – In the former case, we convert the nonstationary series to its stationary counterpart by taking first differences – In the latter case, we convert the nonstationary series to its stationary counterpart by de- trending 12.5 Regression When There is No Cointegration

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Principles of Econometrics, 4t h EditionPage 73 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Consider the random walk model: – This can be rendered stationary by taking the first difference: The variable y t is said to be a first difference stationary series 12.5 Regression When There is No Cointegration 12.5.1 First Difference Stationary

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Principles of Econometrics, 4t h EditionPage 74 Chapter 12: Regression with Time-Series Data: Nonstationary Variables A suitable regression involving only stationary variables is: – Now consider a series y t that behaves like a random walk with drift: with first difference: The variable y t is also said to be a first difference stationary series, even though it is stationary around a constant term 12.5 Regression When There is No Cointegration 12.5.1 First Difference Stationary Eq. 12.11a

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Principles of Econometrics, 4t h EditionPage 75 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Suppose that y and x are I(1) and not cointegrated – An example of a suitable regression equation is: 12.5 Regression When There is No Cointegration 12.5.1 First Difference Stationary Eq. 12.11b

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Principles of Econometrics, 4t h EditionPage 76 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Consider a model with a constant term, a trend term, and a stationary error term: – The variable y t is said to be trend stationary because it can be made stationary by removing the effect of the deterministic (constant and trend) components: 12.5 Regression When There is No Cointegration 12.5.2 Trend Stationary

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Principles of Econometrics, 4t h EditionPage 77 Chapter 12: Regression with Time-Series Data: Nonstationary Variables If y and x are two trend-stationary variables, a possible autoregressive distributed lag model is: 12.5 Regression When There is No Cointegration 12.5.2 Trend Stationary Eq. 12.12

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Principles of Econometrics, 4t h EditionPage 78 Chapter 12: Regression with Time-Series Data: Nonstationary Variables As an alternative to using the de-trended data for estimation, a constant term and a trend term can be included directly in the equation: where: 12.5 Regression When There is No Cointegration 12.5.2 Trend Stationary

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Principles of Econometrics, 4t h EditionPage 79 Chapter 12: Regression with Time-Series Data: Nonstationary Variables If variables are stationary, or I(1) and cointegrated, we can estimate a regression relationship between the levels of those variables without fear of encountering a spurious regression If the variables are I(1) and not cointegrated, we need to estimate a relationship in first differences, with or without the constant term If they are trend stationary, we can either de-trend the series first and then perform regression analysis with the stationary (de-trended) variables or, alternatively, estimate a regression relationship that includes a trend variable 12.5 Regression When There is No Cointegration 12.5.3 Summary

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Principles of Econometrics, 4t h EditionPage 80 Chapter 12: Regression with Time-Series Data: Nonstationary Variables 12.5 Regression When There is No Cointegration 12.5.3 Summary FIGURE 12.4 Regression with time-series data: nonstationary variables

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Principles of Econometrics, 4t h EditionPage 81 Chapter 12: Regression with Time-Series Data: Nonstationary Variables Key Words

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Principles of Econometrics, 4t h EditionPage 82 Chapter 12: Regression with Time-Series Data: Nonstationary Variables autoregressive process cointegration Dickey–Fuller tests difference stationary mean reversion nonstationary Keywords order of integration random walk process random walk with drift spurious regressions stationary stochastic process stochastic trend tau statistic trend stationary unit root tests

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