Economtrics of money and finance Lecture six: spurious regression and cointegration

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Economtrics of money and finance Lecture six: spurious regression and cointegration Zongxin Qian School of Finance, Renmin University of China October 21, 2014

Table of Contents Overview Spurious regression Cointegration Error correction model Application: testing the PPP Application: testing the quantity theory of money

Overview Spurious regression Cointegration Error correction model Application: testing the PPP and quantity theory of money

Table of Contents Overview Spurious regression Cointegration Error correction model Application: testing the PPP Application: testing the quantity theory of money

Spurious regression y t = βx t + u t

Spurious regression y t = βx t + u t ˆβ = T t=1 xtyt T t=1 x2 t

Spurious regression y t = βx t + u t ˆβ = T t=1 xtyt T t=1 x2 t ˆβ = β + 1 T T t=1 xtut 1 T T t=1 x2 t

Spurious regression y t = βx t + u t ˆβ = T t=1 xtyt T t=1 x2 t ˆβ = β + 1 T T t=1 xtut 1 T T t=1 x2 t The second term on the RHS E(xtut) E(xt 2) x t and y t are I(0). = 0 in probability if

Spurious regression y t = βx t + u t ˆβ = T t=1 xtyt T t=1 x2 t ˆβ = β + 1 T T t=1 xtut 1 T T t=1 x2 t The second term on the RHS E(xtut) E(xt 2) x t and y t are I(0). Therefore, ˆβ β in probability = 0 in probability if

Spurious regression However, if x t, y t are I(1) and u t I (1), ˆβ Q, where Q is a continuous random variable.

Spurious regression However, if x t, y t are I(1) and u t I (1), ˆβ Q, where Q is a continuous random variable. Therefore, P( ˆβ = 0) = 0.

Spurious regression However, if x t, y t are I(1) and u t I (1), ˆβ Q, where Q is a continuous random variable. Therefore, P( ˆβ = 0) = 0. β = 0 if x t and y t have no correlation. This is called spurious regression.

Table of Contents Overview Spurious regression Cointegration Error correction model Application: testing the PPP Application: testing the quantity theory of money

Definition of cointegration If x t and y t are I(1), but u t = y t βx t is I(0), we say that x t and y t are cointegrated.

Definition of cointegration If x t and y t are I(1), but u t = y t βx t is I(0), we say that x t and y t are cointegrated. Cointegration means that x t and y t have a stable long-run relationship. Therefore, it is widely used for testing long-run relationship between two variables with stochastic trend.

Definition of cointegration If x t and y t are I(1), but u t = y t βx t is I(0), we say that x t and y t are cointegrated. Cointegration means that x t and y t have a stable long-run relationship. Therefore, it is widely used for testing long-run relationship between two variables with stochastic trend. The concept of cointegration applies to more than two I(1) variables. In general, variables are cointegrated if one of their linear combination is I(0).

A nice property of cointegration We know ˆβ = β + 1 T T t=1 xtut 1 T T t=1 x2 t

A nice property of cointegration We know ˆβ = β + 1 T Suppose T t=1 xtut 1 T T t=1 x2 t u t = ρu t 1 + e t, ρ < 1 = t 1 ρ i e i i=0 x t = x t 1 + w t t 1 = x 0 + i=0 w i e t, w t i.i.d.(0, 1)

A nice property of cointegration We know ˆβ = β + 1 T Suppose T t=1 xtut 1 T T t=1 x2 t u t = ρu t 1 + e t, ρ < 1 = t 1 ρ i e i i=0 x t = x t 1 + w t t 1 = x 0 + i=0 w i e t, w t i.i.d.(0, 1) Assuming e t, w t are independent, it is easy to see that 1 0. T T t=1 xtut 1 T T t=1 x2 t

The Engle-Granger two-step cointegration test First, run the regression y t = α + βx t + u t and obtain the residuals

The Engle-Granger two-step cointegration test First, run the regression y t = α + βx t + u t and obtain the residuals second, apply unit root test to û t

The Engle-Granger two-step cointegration test First, run the regression y t = α + βx t + u t and obtain the residuals second, apply unit root test to û t Note that the ADF test critical values for û t are different from the ADF test critical values for x t and y t. Eviews can calculate those critical values for us. See steps in the next slide.

The Engle-Granger two-step cointegration test in Eviews open two variables, say, x t and y t, as a group

The Engle-Granger two-step cointegration test in Eviews open two variables, say, x t and y t, as a group view/cointegration test/single-equation cointegration test choose Engle-Granger as the test method choose the specification of your first step regression. You can include a constant or time trend into the regression model by playing with options in model specification you can choose lag orders of the test according to various information criteria

Table of Contents Overview Spurious regression Cointegration Error correction model Application: testing the PPP Application: testing the quantity theory of money

The ECM We have learned that the cointegrating equation gives the long-run relationship between x t and y t, let s say u t = y t βx t = 0

The ECM We have learned that the cointegrating equation gives the long-run relationship between x t and y t, let s say u t = y t βx t = 0 In economics, the stable long-run relationship could be an equilibrium state. In the short run, the economy might deviate from the equilibrium state. In other words, u t 0

The ECM We have learned that the cointegrating equation gives the long-run relationship between x t and y t, let s say u t = y t βx t = 0 In economics, the stable long-run relationship could be an equilibrium state. In the short run, the economy might deviate from the equilibrium state. In other words, u t 0 However, in many cases, those short-run deviations will be corrected over time and the economy will go back to the equilibrium state as time goes by.

The ECM We have learned that the cointegrating equation gives the long-run relationship between x t and y t, let s say u t = y t βx t = 0 In economics, the stable long-run relationship could be an equilibrium state. In the short run, the economy might deviate from the equilibrium state. In other words, u t 0 However, in many cases, those short-run deviations will be corrected over time and the economy will go back to the equilibrium state as time goes by. We call the short-run deviations errors, and the restoring of the long-run equilibrium relationship error correction.

The ECM A typical ECM model has the following form y t = a + bu t 1 + c(l) y t 1 + d(l) x t 1 + e t, where e t is a white noise.

The ECM A typical ECM model has the following form y t = a + bu t 1 + c(l) y t 1 + d(l) x t 1 + e t, where e t is a white noise. Note that all variables are I(0) in the ECM. Therefore, the model can be estimated by OLS.

Table of Contents Overview Spurious regression Cointegration Error correction model Application: testing the PPP Application: testing the quantity theory of money

The PPP The law of one price (LOP): without any transaction cost, prices of two identical goods should be the same across countries.

The PPP The law of one price (LOP): without any transaction cost, prices of two identical goods should be the same across countries. Under the following assumptions, we can derive the purchasing power parity (PPP) from the LOP. same consumption basket across countries no non-tradables no trade costs no differences in quality and any other aspects of tradable goods

The PPP E = P/P

The PPP E = P/P logp = log(ep )

The PPP E = P/P logp = log(ep ) The above equation is a long-run equilibrium relationship

The PPP E = P/P logp = log(ep ) The above equation is a long-run equilibrium relationship The PPP holds, if a = 0, b = 1, u t I (0) in logp = a + blog(ep ) + u t (1)

The PPP E = P/P logp = log(ep ) The above equation is a long-run equilibrium relationship The PPP holds, if a = 0, b = 1, u t I (0) in logp = a + blog(ep ) + u t (1) it is actually a test for cointegration between p logp and f log(ep )

Table of Contents Overview Spurious regression Cointegration Error correction model Application: testing the PPP Application: testing the quantity theory of money

The quantity theory of money M = kpy

The quantity theory of money M = kpy In the long run, k is a constant determined by culture and financial institutions, and Y is fixed at its nature level.

The quantity theory of money M = kpy In the long run, k is a constant determined by culture and financial institutions, and Y is fixed at its nature level. Therefore, logm t = c + logp t + u t, where u t = 0 if the quantity theory of money holds. There is a long-run relationship between the price level and the quantity of money.

The quantity theory of money M = kpy In the long run, k is a constant determined by culture and financial institutions, and Y is fixed at its nature level. Therefore, logm t = c + logp t + u t, where u t = 0 if the quantity theory of money holds. There is a long-run relationship between the price level and the quantity of money. There could be short-run deviations from the long-run relationship, so it is adequate that u t I (0).

A Chinese example We use CPI and M2 data from 1997m1 to 2006m12.

A Chinese example We use CPI and M2 data from 1997m1 to 2006m12. Why start the sample from 1997 and end the sample by 2006? Please think about it.

A Chinese example We use CPI and M2 data from 1997m1 to 2006m12. Why start the sample from 1997 and end the sample by 2006? Please think about it. All data are seasonally adjusted by the census X12 method.

Null Hypothesis: M_SA has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic - based on SIC, maxlag=12) t-statistic Prob.* Augmented Dickey-Fuller test statistic 1.484774 1.0000 Test critical values: 1% level -4.036983 5% level -3.448021 10% level -3.149135 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(M_SA) Method: Least Squares Date: 10/20/14 Time: 21:56 Sample (adjusted): 1997M02 2006M12 Included observations: 119 after adjustments Variable Coefficient Std. Error t-statistic Prob. M_SA(-1) 0.007945 0.005351 1.484774 0.1403 C 150.2594 304.1936 0.493960 0.6223 @TREND(1997M01) 11.54156 11.80815 0.977423 0.3304

Null Hypothesis: P_SA has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic - based on SIC, maxlag=12) t-statistic Prob.* Augmented Dickey-Fuller test statistic -0.360908 0.9879 Test critical values: 1% level -4.036983 5% level -3.448021 10% level -3.149135 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(P_SA) Method: Least Squares Date: 10/20/14 Time: 22:12 Sample (adjusted): 1997M02 2006M12 Included observations: 119 after adjustments Variable Coefficient Std. Error t-statistic Prob. P_SA(-1) -0.006161 0.017070-0.360908 0.7188 C 1.835840 5.948662 0.308614 0.7582 @TREND(1997M01) 0.011227 0.005278 2.127144 0.0355 R-squared 0.062082 Mean dependent var 0.280362 Adjusted R-squared 0.045911 S.D. dependent var 1.372156 S.E. of regression 1.340287 Akaike info criterion 3.448531

Date: 10/20/14 Time: 22:13 Series: P_SA M_SA Sample: 1997M01 2006M12 Included observations: 120 Null hypothesis: Series are not cointegrated Cointegrating equation deterministics: C Automatic lags specification based on Schwarz criterion (maxlag=12) Dependent tau-statistic Prob.* z-statistic Prob.* P_SA -1.194623 0.8601-3.097073 0.8710 M_SA -1.769938 0.6459-3.856607 0.8181 *MacKinnon (1996) p-values. Intermediate Results: P_SA M_SA Rho - 1-0.026026-0.032408 Rho S.E. 0.021786 0.018311 Residual variance 1.815733 65274243 Long-run residual variance 1.815733 65274243 Number of lags 0 0 Number of observations 119 119 Number of stochastic trends** 2 2 **Number of stochastic trends in asymptotic distribution

Summary of the Chinese example Both variables are I(1) But the null of no cointegration is not rejected No support for the quantity theory

Summary of the Chinese example Both variables are I(1) But the null of no cointegration is not rejected No support for the quantity theory But the single-equation residual-based test has limitations. 1. It allows for only one cointegration relationship 2. The specification of the first-step regression model might be too simple

Summary of the Chinese example Both variables are I(1) But the null of no cointegration is not rejected No support for the quantity theory But the single-equation residual-based test has limitations. 1. It allows for only one cointegration relationship 2. The specification of the first-step regression model might be too simple We shall introduce another test based on system estimation in a coming lecture

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