Error correction model

ECMerror correction modelserror-correction modelerror/correction
An error correction model (ECM) belongs to a category of multiple time series models most commonly used for data where the underlying variables have a long-run stochastic trend, also known as cointegration.wikipedia
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Cointegration

cointegratedco-integratingco-integration
An error correction model (ECM) belongs to a category of multiple time series models most commonly used for data where the underlying variables have a long-run stochastic trend, also known as cointegration.

Vector autoregression

VARvector autoregressive modelstructural VAR estimation
The resulting model is known as a vector error correction model (VECM), as it adds error correction features to a multi-factor model known as vector autoregression (VAR).

Johansen test

Johansen cointegration testJohansen's method
Among these are the Engle and Granger 2-step approach, estimating their ECM in one step and the vector-based VECM using Johansen's method.
There are two possible specifications for error correction: that is, two vector error correction models (VECM):

Time series

time series analysistime-seriestime-series analysis
An error correction model (ECM) belongs to a category of multiple time series models most commonly used for data where the underlying variables have a long-run stochastic trend, also known as cointegration.

Spurious relationship

spurious correlationspuriousmisleading results
Yule (1936) and Granger and Newbold (1974) were the first to draw attention to the problem of spurious correlation and find solutions on how to address it in time series analysis.

Regression analysis

regressionmultiple regressionregression model
Given two completely unrelated but integrated (non-stationary) time series, the regression analysis of one on the other will tend to produce an apparently statistically significant relationship and thus a researcher might falsely believe to have found evidence of a true relationship between these variables.

Ordinary least squares

OLSleast squaresOrdinary least squares regression
Ordinary least squares will no longer be consistent and commonly used test-statistics will be non-valid. If the regression is not spurious as determined by test criteria described above, Ordinary least squares will not only be valid, but in fact super consistent (Stock, 1987).

Monte Carlo method

Monte CarloMonte Carlo simulationMonte Carlo methods
In particular, Monte Carlo simulations show that one will get a very high R squared, very high individual t-statistic and a low Durbin–Watson statistic.

Coefficient of determination

R-squaredR'' 2 R 2
In particular, Monte Carlo simulations show that one will get a very high R squared, very high individual t-statistic and a low Durbin–Watson statistic.

T-statistic

Student's t-statistict''-statisticStudent's ''t''-statistic
In particular, Monte Carlo simulations show that one will get a very high R squared, very high individual t-statistic and a low Durbin–Watson statistic.

Durbin–Watson statistic

Durbin–Watson testDurbin–Watsonautocorrelated residuals
In particular, Monte Carlo simulations show that one will get a very high R squared, very high individual t-statistic and a low Durbin–Watson statistic.

Convergence of random variables

convergence in distributionconverges in distributionconvergence in probability
Technically speaking, Phillips (1986) proved that parameter estimates will not converge in probability, the intercept will diverge and the slope will have a non-degenerate distribution as the sample size increases.

Y-intercept

intercepty''-interceptintercepts
Technically speaking, Phillips (1986) proved that parameter estimates will not converge in probability, the intercept will diverge and the slope will have a non-degenerate distribution as the sample size increases.

Trend stationary

trendtrend-stationaritytrend-stationary
Because of the stochastic nature of the trend it is not possible to break up integrated series into a deterministic (predictable) trend and a stationary series containing deviations from trend.

Random walk

random walkssimple random walkRandom walks on graphs
Even in deterministically detrended random walks spurious correlations will eventually emerge.

Box–Jenkins method

Box–Jenkins Box–Jenkins approachBox–Jenkins analysis
In order to still use the Box–Jenkins approach, one could difference the series and then estimate models such as ARIMA, given that many commonly used time series (e.g. in economics) appear to be stationary in first differences.

Autoregressive integrated moving average

ARIMAAutoregressive integrated moving average modelAutoregressive integrated moving average (ARIMA)
In order to still use the Box–Jenkins approach, one could difference the series and then estimate models such as ARIMA, given that many commonly used time series (e.g. in economics) appear to be stationary in first differences.

Denis Sargan

John Denis SarganJohn D. SarganSargan
This led Sargan (1964) to develop the ECM methodology, which retains the level information.

Stationary process

stationarynon-stationarystationarity
The first step of this method is to pretest the individual time series one uses in order to confirm that they are non-stationary in the first place.

Unit root

difference stationarynon-stationary
This can be done by standard unit root DF testing and ADF test (to resolve the problem of serially correlated errors).

Augmented Dickey–Fuller test

ADF testaugmentedaugmented Dickey–Fuller
This can be done by standard unit root DF testing and ADF test (to resolve the problem of serially correlated errors).

Consistent estimator

consistentconsistencyinconsistent
If the regression is not spurious as determined by test criteria described above, Ordinary least squares will not only be valid, but in fact super consistent (Stock, 1987).

Permanent income hypothesis

permanent incomepermanent income theory
Suppose, consumption C_t and disposable income Y_t are macroeconomic time series that are related in the long run (see Permanent income hypothesis).