# Cointegration and present value models

Cointegration is an important tool for modeling the long-run relationships in time series data. If you work with time series data, you will likely find yourself needing to use cointegration at some point.

This blog provides an in-depth introduction to cointegration and will cover all the nuts and bolts you need to get started. In particular, we will look at:. Though not necessary, you may find it helpful to review the blogs on time series modeling and unit root testing before continuing with this blog.

Economic theory suggests that many time series datasets will move together, fluctuating around a long-run equilibrium. In econometrics and statistics, this long-run equilibrium is tested and measured using the concept of cointegration. Cointegration occurs when two or more nonstationary time series :. All three series are nonstationary time series. This vector:. Normalization Because there can be multiple cointegrating vectors that fit the same economic model, we must impose identification restrictions to normalize the cointegrating vector for estimation.

## Fractional cointegration and tests of present value models

For example, applying these restrictions to our earlier system yields. Part of the appeal of this normalization is that it can be rewritten in a standard regression form. While the regression format is a common normalization, it is important to remember that economic theory should inform our identifying restrictions.

Cointegration implies that time series will be connecting through an error correction model. The error correction model is important in time series analysis because it allows us to better understand long-run dynamics. Additionally, failing to properly model cointegrated variables can result in biased estimates. The error correction model depicts the dynamics of a variable as a function of the deviations from long-run equilibrium. If the cointegrating vector has been previously estimated, then standard OLS or DOLS can be used to estimate the error correction relationship.

In this case:. If we are working in a vector autoregressive context, cointegration implies a VECM such that. Before jumping directly to cointegration testing, there are a number of other time series modeling modeling steps that we should consider first. One of the key considerations prior to testing for cointegration, is whether there is theoretical support for the cointegrating relationship. It is important to remember that cointegration occurs when separate time series share an underlying stochastic trend.

The idea of a shared trend should be supported by economic theory.We propose a test for stochastic cointegration against the alternative of no cointegration and a secondary test for stationary cointegration against the heteroscedastic alternative.

Asymptotic distributions of these tests under their respective null hypotheses are derived and consistency under their respective alternatives is established. In contrast to conventional cointegration tests, which we show via simulation are unreliable in the presence of the kind of volatility typical of financial data, our tests are able to uncover new cointegration evidence in favour of the present value model, particularly in the bond market.

Location of Repository. Provided by: Research Papers in Economics. Suggested articles. Citations A cointegration approach to estimating preference parameters, A general method of testing for random parameter variation in statistical models A residual based test of the null of cointegration against the alternative of no cointegration. A simple procedure for detecting periodically collapsing rational bubbles, An introduction to stochastic unit root processes, Cointegration and error correction: representation, estimation and testing, Cointegration and tests of present value models, Convergence to stochastic integrals for dependent heterogeneous processes, Earnings and expected returns, Explaining stock price movements: is there a case for fundamentals?

Federal Reserve Bank of Dallas Review, Explosive rational bubbles in stock prices, Heteroscedastic cointegration, Heteroskedasticity and autocorrelation consistent covariance matrix estimation, Interpreting tests of the convergence hypothesis, Intrinsic bubbles: The case of stock prices, Low frequency movements in stock prices: a state-space decomposition, Review of Economics and Statistics, Stochastic cointegration: estimation and inference, Testing the null of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root?

The second is that, despite this trending behaviour, such series often tend to co-move over time according to a stationary, or I 0process i. Many empirical tests of important theoretical economic hypotheses are carried out within the EG cointegration framework. Two noteworthy examples are the relationship between long run and short run interest rates and the relationship between dividends and stock prices, which belong to the class of hypotheses known as present value models PVM ; see Campbell and Shiller That is, the individual series often appear visually to be more volatile, or less smooth, than would be consistent with I 1 behaviour, particularly when using higher frequency data.

An example of this type of volatility and co-movement is given in Figure 1. The story relating dividends and prices in the US stock market is similar. This approach permits a much richer range of behaviour than is possible under EG.

Given that the HMLa framework seems to be an appropriate means of analysis for volatile data, in Section 3 we turn to the issue of hypothesis testing in a regression model representation. The central hypothesis of interest is whether series are stochastically cointegrated either stationary or heteroscedasticor not cointegrated. This parallels the EG approach and we suggest a statistic to test the null of stochastic cointegration based on regression residuals.

Within stochastic cointegration, we also consider the hypothesis that the cointegration is stationary against the alternative that it is heteroscedastic and we suggest a second residual-based statistic to test this. Both statistics have the advantage of being very simple to construct.

Conveniently, both are shown to have straightforward normal limiting distributions that, unlike most cointegration tests, do not depend on the number of regressors involved. Their consistency properties under associated alternative hypotheses are also established. Finally, in Section 6 we give a detailed stochastic cointegration analysis of the evidence for and against the cointegration implications of the PVM in the bond and stock market.

For the US stock market, evidence supporting the PVM is also found, contingent on the time period under study. Interestingly, for all the series we consider here, we conclude they are better modelled by heteroscedastically integrated rather than I 1 processes.

Only the process zt is observed. So, a stochastically integrated variable encompasses both ordinary and heteroscedastic integration. Because of the stationary behaviour of c0 zt in either case, we simply refer to this as stationary cointegration.

Vector Error Correction Model (VECM) - Step 4 of 4

Thus, stochastic cointegration encompasses both stationary cointegration possibly of the EG kind and heteroscedastic cointegration. The interest spread is shown in Figure 4 and this shows little visual evidence of a stochastic trend, whereas heteroscedasticity remains a distinct possibility.

We will see in Section 6 that our new tests indicate that these two series are in fact heteroscedastically cointegrated. To do this we need to be more precise about the statistical properties of the disturbances in the model and so we make extensive use of the following linear process assumption in the remainder of the paper.

### Cointegration and Tests of Present Value Models

Assumption LP. Well-known examples of stochastically trendless processes are weakly stationary series such as those in Assumption LP this includes the stationary AR 1 case as used in the discussion below. The central example of a process with a stochastic trend is an I 1 process such as wt. It is this property that bestows meaning to the concept of co-movement of a heteroscedastic kind.

The proof of the proposition is somewhat tedious under the generality of Assumption LP, but a simple example illustrates the result. This shows that the stochastically trendless property of vt dominates the multiplicative process vt wt.

Within stochastic cointegration, we may wish to know whether stationary or heteroscedastic cointegration pertains. The rank condition ensures that further sub-relationships among the xt variables in 7 are excluded.National Bureau of Economic Research.

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### Cointegration and Tests of Present Value Models

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Other versions of this item: John Y. Shiller, John Y. Shapiro, Sargent, Shiller, Robert J. Robert J. Gregory Mankiw, N. Matthew D. Gregory Mankiw, Marsh, Terry A. Shiller, Robert J, Miron, Campbell, John Y, Campbell, Schoenholtz, Sargent, Thomas J.

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John Y. Shiller, Shapiro, Sargent, Shiller, Robert J. Robert J. Gregory Mankiw, N. Matthew D. Gregory Mankiw, Marsh, Terry A. Shiller, Robert J, Miron, Campbell, John Y, Campbell, Schoenholtz, Sargent, Thomas J. Thomas J. Allan W. Veall, Ito, Takatoshi, Takatoshi Ito, Scott, Louis O, Hamilton, James D.

Kleidon, Allan W, Sawa, Takamitsu, Full references including those not matched with items on IDEAS More about this item Keywords Cointegration ; present value methods ; stock price index ; interest rates ; term structure ; volatility ; efficient markets ; Statistics Access and download statistics Corrections All material on this site has been provided by the respective publishers and authors.

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