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1 MODELS WITH A LAGGED DEPENDENT VARIABLE A widely-used solution to the problem of including dynamics in a model while mitigating the problem of multicollinearity is to employ an autoregressive distributed lag model, often written ADL(p, q). ADL(p, q)

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2 The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent variable are included on the right side as explanatory variables. MODELS WITH A LAGGED DEPENDENT VARIABLE ADL(p, q) ADL = autoregressive distributed lag 'autoregressive' because Y t depends on previous values of Y.

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3 p is the maximum number of lags of the dependent variable used in this way. q is the maximum lag of the X variable, or variables if there are several. MODELS WITH A LAGGED DEPENDENT VARIABLE ADL(p, q) ADL = autoregressive distributed lag 'autoregressive' because Y t depends on previous values of Y. p = maximum number of lags of the dependent variable q = maximum number of lags of the X variable(s)

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4 The ADL model is particularly appealing when the dependent variable exhibits a high degree of dependence because then, as a matter of common sense, its value in one observation is likely to be influenced by its value in the previous one. MODELS WITH A LAGGED DEPENDENT VARIABLE ADL(p, q) ADL = autoregressive distributed lag 'autoregressive' because Y t depends on previous values of Y. p = maximum number of lags of the dependent variable q = maximum number of lags of the X variable(s)

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5 It is econometrically attractive because it can accommodate a broad range of dynamic patterns with relatively few lag terms and parameters. (It is parsimonious, to use the technical term.) MODELS WITH A LAGGED DEPENDENT VARIABLE ADL(p, q) ADL = autoregressive distributed lag 'autoregressive' because Y t depends on previous values of Y. p = maximum number of lags of the dependent variable q = maximum number of lags of the X variable(s)

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6 This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity. MODELS WITH A LAGGED DEPENDENT VARIABLE ADL(p, q) ADL = autoregressive distributed lag 'autoregressive' because Y t depends on previous values of Y. p = maximum number of lags of the dependent variable q = maximum number of lags of the X variable(s)

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7 We will start with the simplest model of all, the ADL(1,0) model where the only lagged variable is the lagged dependent variable. MODELS WITH A LAGGED DEPENDENT VARIABLE ADL(1, 0) ADL = autoregressive distributed lag 'autoregressive' because Y t depends on previous values of Y. p = 1 in this specification q = 0

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8 Given the continuity of many time series processes, Y t–1, the value of a time series at time t – 1, is often the most important determinant of its value Y t at time t. and it makes sense to include it explicitly in the model as an explanatory variable. MODELS WITH A LAGGED DEPENDENT VARIABLE ADL = autoregressive distributed lag 'autoregressive' because Y t depends on previous values of Y. p = 1 in this specification q = 0 ADL(1, 0)

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9 We will begin by investigating the dynamics implicit in the model graphically. We will suppose, for convenience, that 2 is positive and that X increases with time, and we will neglect the effect of the disturbance term. MODELS WITH A LAGGED DEPENDENT VARIABLE 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t

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10 We shall suppose throughout this section that │ 3 │ < 1. This is a stability condition for the process. We will discuss the consequences of violations of this condition in Chapter 13. We will in fact assume 0 < 3 < 1 because Y t and Y t–1 are typically positively correlated. MODELS WITH A LAGGED DEPENDENT VARIABLE 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t

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11 At time t, Y t is given by the equation at the top. It is represented by the point corresponding to X t on the lowest of the five lines in the figure. Y t–1 has already been determined at time t, so the term 3 Y t–1 is fixed. MODELS WITH A LAGGED DEPENDENT VARIABLE 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t

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12 The equation thus may be viewed as giving the short-run relationship between Y t and X t for period t. ( 1 + 3 Y t–1 ) is effectively the intercept and 2, the slope coefficient, gives the short- run effect of X on Y. MODELS WITH A LAGGED DEPENDENT VARIABLE 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t

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13 When we come to time t + 1, Y t+1 is given by the second equation and the effective intercept is now ( 1 + 3 Y t ). Since X is increasing, Y is increasing, so the intercept is larger than that for Y t and the short-run relationship has shifted upwards. The slope is unchanged, 2. MODELS WITH A LAGGED DEPENDENT VARIABLE 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t

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14 Thus two factors are responsible for the growth of Y over time: the direct effect of the increase in X, and the gradual upward shift of the short-run relationship. The figure shows the outcomes for time t as far as time t + 4. MODELS WITH A LAGGED DEPENDENT VARIABLE 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t

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15 You can see that the actual relationship between Y and X, traced out by the markers representing the observations, is steeper than the short-run relationship for each time period. MODELS WITH A LAGGED DEPENDENT VARIABLE 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t

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16 We will determine the long-run relationship between Y and X by performing a comparative statics analysis described in the previous slideshow. MODELS WITH A LAGGED DEPENDENT VARIABLE 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t

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17 Denoting equilibrium Y and X by Y and X, Y t = Y t–1 = Y and X t = X in equilibrium. Hence, ignoring the transient effect of the disturbance term, the equilibrium relationship is as shown above. 1 + 3 Y t+3 XtXt X t+1 X t+3 X t+4 X Y X t+2 1 + 3 Y t+2 1 + 3 Y t+1 1 + 3 Y t 1 + 3 Y t–1 short-run relationship at time t MODELS WITH A LAGGED DEPENDENT VARIABLE

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18 Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The factor 2 / (1 – 3 ) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We will describe this as the long-run effect. MODELS WITH A LAGGED DEPENDENT VARIABLE long-run effect of X on Y

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19 In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie between 0 and 1. MODELS WITH A LAGGED DEPENDENT VARIABLE long-run effect of X on Y if

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20 Another way of exploring the dynamics is to look at the implicit relationship between Y t and current and lagged values of X. If the relationship is true for time period t, it is also true for time period t – 1. MODELS WITH A LAGGED DEPENDENT VARIABLE

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21 We substitute for Y t–1 in the first equation. MODELS WITH A LAGGED DEPENDENT VARIABLE

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22 Continuing to lag and substitute, one obtains the equation shown. MODELS WITH A LAGGED DEPENDENT VARIABLE

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23 Hence Y t can be viewed as a linear combination of current and lagged values of X with a lag distribution that consists of geometrically declining weights: 2, 2 3, 2 3 2,... MODELS WITH A LAGGED DEPENDENT VARIABLE

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24 We have thus found a way of allowing lagged values of X to influence Y without introducing them into the model explicitly and giving rise to multicollinearity. MODELS WITH A LAGGED DEPENDENT VARIABLE

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25 One should, however, note that this particular pattern of weights, known as a Koyck lag distribution, embodies the assumption that more recent values of X have more influence than older ones, and that the rate of decline in the weights is constant. MODELS WITH A LAGGED DEPENDENT VARIABLE Koyck lag distribution: geometrically declining weights

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26 We will see in due course that we can relax both of these constraints. MODELS WITH A LAGGED DEPENDENT VARIABLE

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27 From the last equation it can be seen that, at time t, with X t–1, X t–2, etc already determined, the only influence of X on Y is via X t. For this reason, again, we describe 2 as the short-run effect. MODELS WITH A LAGGED DEPENDENT VARIABLE short-run effect of X on Y

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28 This representation of the model also yields the same long-run effect, as before. The proof is left as an exercise. MODELS WITH A LAGGED DEPENDENT VARIABLE

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Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 11.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics

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