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Cointegration and Error Correction Model

This part discusses a new theory for a regression with nonstationary unit root variables. In general, this should require a di¤erent treatment from a conventional regression with stationary variables, which has been covered so far. In particular, we focus on a class of the linear combination of the unit root processes known as cointegrated process.

1.1

Stylized Facts about Economic Time Series

Casual inspection of most economic time series data such as GNP and prices reveals that these series are non stationary. We can characterize some of the key feature of the various series as follows:

1. Most of the series contain a clear trend. In general, it is hard to distinguish between trend stationary and di¤erence stationary processes. 2. Some series seem to meander. For example, the pound/dollar exchange rate shows no particular tendency to increase or decrease. The pound seems to go through sustained periods of appreciation and then depreciation with no tendency to revert to a long-run mean. This type of random walk behavior is typical of unit root series.

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3. Any shock to a series displays a high degree of persistence. For example, the UK industrial production plummeted in the late 1970s and not returning to its previous level until mid 80s. Overall the general consensus is at least empirically that most macro economic time series follow a unit root process.

4 Some series share co-movements with other series. For example, shortand long-term interest rate, though meandering individually, track each other quite closely maybe due to the underlying common economic forces. This phenomenon is called cointegration.

A note on notations: It is widely used that the unit root process is called an integrated of order 1 or for short I (1) process. On the other hand, a stationary process is called an I (0) process.1

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In this regard we can de…ne I (d) process, and d is a number of di¤erencing to render the series stationary.

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1.2

Spurious Regression

Suppose that two I (1) processes, yt and xt , are independently distributed. We now consider the following simple regression:

yt = xt + error:

Clearly, there should be no systematic relationship between y and x, and therefore, we should expect that an OLS estimate of

should be close to zero,

or insigni…cantly di¤erent from zero, at least as the sample size increases. But, as will be shown below, this is not the case. This phenomenon originated from Yule (1926) was called “a nonsense correlation.”

Example 1 There are some famous examples for spurious correlation. One is that of Yule (1926, Journal of the Royal Statistical Society), reporting a correlation of 0.95 between the proportion of Church of England marriages to all marriages and the mortality rate over the period 1866-1911. Yet another is the econometric example reported by Hendry (1980, Economica) between the price level and the cumulative rainfall in the UK.2

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This relation proved resilient to many econometric diagnostic tests and was humorously advanced as a new theory of in‡ation.

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As we have come to understand in recent years, it is commonality of (stochastic) trending mechanisms in data that often leads to these spurious relations. What makes the phenomenon dramatic is that it occurs even when the data are otherwise independent.

In a prototypical spurious regression the …tted coe¢ cients are statistically signi…cant when there is no true relationship between the dependent variable and the regressors. Using Monte Carlo simulations Granger and Newbold (1974, Journal of Econometrics ) showed this phenomenon. Phillips (1986, Journal of Econometrics ) derived an analytic proof. These results are summarized in the following theorem:

Theorem 1 (Spurious Regression) Suppose that y and x are independent I (1) variables generated respectively by

yt = yt

iid (0;

2
")

and et

+ "t ;

xt = xt

where "t

1

1

+ et ;

iid (0;

4

2
e ),

and "t and et are independent of

each other. Consider the regression,

(1)

y t = x t + ut :

Then, as T ! 1,
(a) The OLS estimator of

obtained from (1), denoted ^ , does not converge

to (true value of) zero.
(b) The t-statistic testing

= 0 in (1) diverges to

in…nity.

In sum, in the case of spurious regression, ^ takes any value randomly, and its t-statistic always indicates signi…cance of the estimate. Though a formal testing procedure will be needed to detect evidence of the spurious regression or cointegration (see below), one useful guideline is that we are likely to face with the spurious relation when we …nd a highly signi…cant t-ratio combined with a rather low value of R2 and a low value of the Durbin-Watson statistic.

1.3

Cointegration

Economic theory often suggests that certain subset of variables should be linked by a long-run equilibrium relationship. Although the variables under 5

consideration may drift away from equilibrium for a while, economic forces or government actions may be expected to restore equilibrium. Example 2 Consider the market for tomatoes in two parts of a country, the north and the south with prices pnt and pst respectively. If these prices are equal the market will be in equilibrium. So pnt = pst is called an attractor. If the prices are unequal it will be possible to make a pro…t by buying tomatoes in one region and selling them in the other. This trading mechanism will be inclined to equate prices again, raising prices in the buying region and lowering them in selling region.

When the concept of equilibrium is applied to I (1) variables, cointegration occurs; that is, cointegration is de…ned as a certain stationary linear combination of multiple I (1) variables.
Example 3 Consider the consumption spending model. Although both consumption and income exhibit a unit root, over the long run consumption tends to be a roughly constant proportion of income, so that the di¤erence between the log of consumption and log of income appears to be a stationary process. Example 4 Another well-known example is the theory of Purchasing Power Parity (PPP). This theory holds that apart from transportation costs, goods 6

should sell for the same e¤ective price in two countries. Let Pt denote the index of the price level in US (in dollars per good), Pt denote the price index for UK (in pounds per good), and St the rate of exchange between the currencies (in dollars per pound). Then the PPP holds that Pt = St Pt ; taking logarithm, pt = st + pt ; where pt = ln(Pt ); st = ln(St ); pt = ln(Pt ): In practice, errors in measuring prices, transportation costs and di¤erences in quality prevent PPP from holding exactly at every date t: A weaker version of the hypothesis is that the variable zt de…ned by zt = pt

st

pt is stationary,

even though the individual elements (pt ; st ; pt ) are all I (1). Cointegration brings with it two obvious econometric questions. The …rst is how to estimate the cointegrating parameters and the second is how to test whether two or more variables are cointegrated or spurious.

We …rst examine estimation of the cointegrating regression. Consider the simple time series regression,3

yt = x t + ut ;
3

(2)

The deterministic regressors such as intercept and a linear time trend can be easily accommodated in the regression without changing the results in what follows.

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where xt is an I (1) variable given by

xt = xt

1

(3)

+ et :

Since xt is I (1), it follows that yt is I (1). But, for yt and xt to be cointegrated...