On substituting the expression for permanent income

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On substituting the expression for permanent income into the equation of the consumption function, we get (59) y ( t ) = γ (1 φ ) 1 φL x ( t ) + ε ( t ) . When the notation γ (1 φ ) = β is adopted, equation (59) becomes identical to the equation (45) of the Koyk model. 22
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EC3062 ECONOMETRICS Error-Correction Forms, and Nonstationary Signals The usual linear regression procedures presuppose that the relevant moment matrices will converge asymptotically to fixed limits as the sample size increases. This cannot happen if the data are trended, in which case, the standard techniques of statistical inference will not be applicable. A common approach is to subject the data to as many differencing operations as may be required to achieve stationarity. However, differencing tends to remove some of the essential information regarding the behaviour of economic agents. Moreover, it is often discovered that the regression model looses much of its explanatory power when the differences of the data are used instead. In such circumstances, one might use the so-called error-correction model. The model depicts a mechanism whereby two trended economic variables maintain an enduring long-term proportionality with each other. The data sequences comprised by the model are stationary, either in- dividually or in an appropriate combination; and this enables us apply the standard procedures of statistical inference that are appropriate to models comprising data from stationary processes. 23
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EC3062 ECONOMETRICS Consider taking y ( t 1) from both sides of the equation of (51) which represents the first-order dynamic model. This gives (60) y ( t ) = y ( t ) y ( t 1) = ( φ 1) y ( t 1) + βx ( t ) + ε ( t ) = (1 φ ) β 1 φ x ( t ) y ( t 1) + ε ( t ) = λ γx ( t ) y ( t 1) + ε ( t ) , where λ = 1 φ and where γ is the gain of the transfer function as defined under (41). This is the so-called error-correction form of the equation; and it indicates that the change in y ( t ) is a function of the extent to which the proportions of the series x ( t ) and y ( t 1) differs from those which would prevail in the steady state. The error-correction form provides the basis for estimating the param- eters of the model when the signal series x ( t ) is trended or nonstationary. A pair of nonstationary series that maintain a long-run proportionality are said to be cointegrated. It is easy to obtain an accurate estimate of γ , which is the coefficient of proportionality, simply by running a regression of y ( t 1) on x ( t ). 24
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EC3062 ECONOMETRICS To see how to derive an error-correction form for a more general autore- gressive distributed-lag model, consider the second-order model: (61) y ( t ) = φ 1 y ( t 1) + φ 2 y ( t 2) + β 0 x ( t ) + β 1 x ( t 1) + ε ( t ) . The part φ 1 y ( t 1)+ φ 2 y ( t 2) comprising the lagged dependent variables can be reparameterised as follows: [ φ 1 φ 2 ] 1 0 1 1 1 0 1 1 y ( t 1) y ( t 2)) = [ θ ρ ] y ( t 1) y ( t 1)) .
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