Lecture 7 Applied Finance PDF

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Applied Finance with Eviews– ECOM122

Lecture 7: Modelling long-term relationships Part I

Lecture 7 Overview

1. Modelling long-run relationships: • Cointegration • Error Correction Model (ECM) : ECM t test of Cointegration 2. ECM estimations (Next week) • Unrestricted ECM estimation • Engle-Granger Two-step Method • Engle-Yoo Three-step Method

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Modelling long-term relationships

Problem 1: What is spurious regression? Regression analysis suggests a relationship exists between variables when no causal relationship exists. The regression is meaningless. This is a nonsense or spurious regression. – A classic example is the very strong positive, non-linear relationship between inflation and cumulative rainfall for the UK, 1958-1978, R2=.998 in Hendry (1980.) – Check: http://www.tylervigen.com/spurious-correlations • The assumptions of the classical linear regression model require the series to be stationary and the disturbances to have zero mean, constant (finite) variance, etc… • We have a problem if 𝑥𝑡 and 𝑦𝑡 are I(1), because, in general, the residuals are also I(1 ). 3

Modelling long-term relationships

• Problem 2: It is not enough to take the first differences, we cannot use ARMA models: Δ𝑦𝑡 = 𝛽Δ𝑥𝑡 + 𝑢𝑡 Because in the long-run, at equilibrium, 𝑦𝑡 = 𝑦𝑡−1 = 𝑦 and 𝑥𝑡 = 𝑥𝑡−1 = 𝑥

=> So all the differences are zero, there is no long-run solution

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Engle-Granger Cointegration

• Formally, suppose to have the following model: 𝑦𝑡 = 𝛽1 + 𝛽2 𝑥𝑡 + 𝑢𝑡 where 𝑦𝑡 and 𝑥𝑡 are integrated series of order 1 (independent random walks): 𝑦𝑡 = 𝑦𝑡−1 + 𝜀1𝑡 where 𝜀1𝑡 ~ 0,1

𝑥𝑡 = 𝑥𝑡−1 + 𝜀2𝑡 where 𝜀2𝑡 ~ 0,1

• After estimating the equation, we can extract the residuals (we can do this easily by clicking on Proc/Make Residual Series from the estimation result window): 𝑢𝑡 = 𝑦𝑡 − 𝛽1 − 𝛽2 𝑥𝑡 • Suppose that we now perform a unit root test on 𝑢𝑡 and we find that it is stationary, I (0). If this is the case, the regression which we carry out on a model like the one in equation above is meaningful. The two variables are cointegrated (Engle and Granger, 1987). 5

Engle-Granger Cointegration

• From an economic point of view, two variables will be cointegrated if they have a long-term, or equilibrium, relationship. Examples in finance: – Spot and future prices for a given commodity or asset – Ratio of relative prices and an exchange rate – Equity prices and dividend

• To sum up if we check the residuals of a regression model and those are I (0), even though the regressand and the regressors are non stationary, the traditional regression methodology is applicable to data involving time series. • Note that you should not be able to run a cointegration test if one variable is stationary and the other is non-stationary. • Using Granger (1986) words: “A test for cointegration can be thought of as a pre-test to avoid 'spurious regression' situations”. 6

Cointegrating Regression Augmented Dickey-Fuller (CRADF) test.

• Let’s perform a unit root test on the residulas: Δ𝑒𝑡 = 𝛼𝑒𝑡−1 + 𝛿1 Δ𝑒𝑡−1 + 𝛿1 Δ𝑒𝑡−1 + ⋯ + 𝛿𝑚 Δ𝑒𝑡−𝑚 + 𝑢𝑡 we use Δ𝑒 terms to eliminate any autocorrelation so that 𝑢𝑡 ~ i.i.d. (0, 𝜎 2 ) – Notice there is no constant in the regression. A constant can be included in either the cointegrating regression or the CRADF regression but not both. Most researchers include a constant in the cointegrating regression. – With a constant in the cointegrating regression, the residuals have zero mean. We do not expect the residuals to have a deterministic trend and so a linear trend is not included.

H0 : 𝛼 = 0 and the 𝑒𝑡 are I(1), x and y are not cointegrated H1 : 𝛼 < 0 and the 𝑒𝑡 are I(0), x and y are cointegrated 7

Cointegrating Regression Augmented Dickey-Fuller (CRADF) test.

• Select both series, open them as a group: • Select View/Cointegration Test/Single-Equation Cointegration Test...

• •

Test method: Engel Granger Select the lag specification (information criterion) and the equation trend specification (normally, Constant [level]), additional trends (normally, None). 8

Cointegrating Regression Augmented Dickey-Fuller (CRADF) test.

• The test statistic t has a non-standard distribution. If the calculated value of the test statistic is less than the critical value then the null hypothesis of no cointegration is rejected; the series 𝑥𝑡 and 𝑦𝑡 are cointegrated. The null hypothesis of no cointegration is rejected. – m, the number of lagged Δ𝑒 is selected in the same way as for unit root tests. – MacKinnon critical values are widely used.

• Repeat the process by estimating the reverse cointegrating regression 𝑥𝑡 = 𝛿1 + 𝛿2 𝑦𝑡 + 𝜐𝑡 and carrying out a CRADF test • Asymptotically, the cointegration tests for the forward and reverse regressions are equivalent. In practice, they can give different results. 9

Example of cointegrated series

• The following example uses weekly data for the period beginning on 5 January 2000 and ending on 15 December 2010 (572 observations) for the logarithms of the Budapest Stock Exchange BUX index, LHNS, and BUX futures, LHNF. • We expect this two series to be cointegrated given that market forces arising from no-arbitrage conditions suggest that there should be an equilibrium relationship between them. Spot and future prices may be expected to be cointegrated since they are obviously prices for the same asset at different points in time, and hence will be affected in very similar ways by given pieces of information.

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Example of cointegrated series 30,000

25,000

20,000

15,000

10,000

5,000 00

01

02

03

04

05 HNF

06

07

08

09

10

HNS

• The forward and reverse cointegrating regressions are estimated: Is lhns c lhnf Is lhnf c lhns • We obtain: 11

Example of cointegrated series

• The estimate of the long-run elasticity of stock prices with respect to futures prices is 1.0045. 12

Example of cointegrated series

• This is the reverse equation.

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Example of cointegrated series: CRADF test

• The Engle-Granger CRADF test, assuming that the series are known to be I(1) is:

• The test statistics for the forward and reverse cointegrating regressions are 17.66270 and -17.65561 respectively; both p-values are 0.0000. The null hypothesis of no cointegration is rejected. 14

Error Correction Model (Equilibrium Correction Model)

• Now that we have detected a long-run relationship, we need a better model: Δ𝑦𝑡 = 𝛽1 + 𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − 𝛾1 − 𝛾2 𝑥𝑡−1 + 𝜀𝑡 Where: 𝛽2 = short-run elasticity between x and y 𝛽3 = speed of adjustment (back to equilibrium) 𝛾2 = long-run elasticity between x and y 𝑦𝑡−1 − 𝛾1 − 𝛾2 𝑥𝑡−1 = error correction term y changes as a result of changes in value of the explanatory variable x, and also in part to correct to any disequilibrium that existed during the previous period. 15

Error Correction Model (Equilibrium Correction Model)

• The model is : Δ𝑦𝑡 = 𝛽1 + 𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − 𝛾1 − 𝛾2 𝑥𝑡−1 + 𝜀𝑡 • It can be written as: Δ𝑦𝑡 = 𝛽1∗ + 𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − 𝑥𝑡−1 + 𝛽4 𝑥𝑡−1 + 𝜀𝑡 where 𝛽1∗ = 𝛽1 − 𝛽3 𝛾1 and

• We have

𝛽4 = 𝛽3 (1 − 𝛾2 )

H0: 𝛽3 = 0, no cointegration, against H1: 𝛽3 < 0, cointegration

The test statistic is the t statistic on the error correction term, 𝛽3 . But, The distribution of the test statistic under the null hypothesis is non-standard.

• What does 𝛽4 tell us? 16

Error Correction Model (Equilibrium Correction Model)

• What is the interpretation of the test: H0: 𝛽4 = 0, H1: 𝛽4 ≠ 0, • Let’s see. If 𝛽4 =0 the ECM becomes:

Δ𝑦𝑡 = 𝛽1∗ + 𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − 𝑥𝑡−1 + 𝜀𝑡

• If 𝛽4 ≠ 0 the ECM can be written as:

Δ𝑦𝑡 = 𝛽1∗ + 𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − 𝑥𝑡−1 + 𝛽4 𝑥𝑡−1 + 𝜀𝑡 = = 𝛽1∗ + 𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − 𝛽3 𝑥𝑡−1 + 𝛽4 𝑥𝑡−1 + 𝜀𝑡 = = 𝛽1∗ +𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − [𝛽3 −𝛽4 ]𝑥𝑡−1 + 𝜀𝑡 𝛽3 − 𝛽4 ∴ Δ𝑦𝑡 = 𝛽1∗ + 𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − 𝑥𝑡−1 + 𝜀𝑡 𝛽3 With both x and y in logarithms it is a test of the null hypothesis that the long run elasticity is one. 17

ECM t test of Cointegration

• The relationship between cointegration and ECMs gives rise to a further test of the null hypothesis of no cointegration based on the ECM t statistic. • Consider the basic ECM: Δ𝑦𝑡 = 𝛽1 + 𝛽2 Δ𝑥𝑡 + 𝛽3 𝑦𝑡−1 − 𝑥𝑡−1 + 𝛽4 𝑥𝑡−1 + 𝜀𝑡 • If x and y are cointegrated then the error-correction term enters the model and 𝛽3 < 0. If x and y are not cointegrated then 𝛽3 = 0 • The reverse ECM t test focuses on 'the other' ECM: Δ𝑥𝑡 = 𝛽1 + 𝛽2 Δ𝑦𝑡 + 𝛽3 𝑥𝑡−1 − 𝑦𝑡−1 + 𝛽4 𝑦𝑡−1 + 𝜀𝑡 18

Example of ECM test of cointegration

• Enter the regression in the Command Window: Is D(LHNS) c D(LHNF) (LHNS(-1)-LHNF(-1)) LHNF(-1)

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Example of ECM test of cointegration

• We know that DW is not good when we have lags, so with weekly data, an AR(5) test of autocorrelation produced the following test results:

• The null hypothesis of no autocorrelation is not rejected. • For the ECM t test, the calculated value of the test statistic is -17.53753, p-value 0. The null hypothesis of no cointegration is rejected.

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Example of ECM test of cointegration

• If we repeat with the other ECM t test:

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