Dynamic Econometric Models:A. Autoregressive Model:

Yt = + 0Xt +1Yt-1 +2Yt-2 +kYt-k + et(With lagged dependent variable(s) on the RHS) B. Distributed-lag Model:

Yt = + 0Xt +1Xt-1 +2Xt-2 + + kXt-k + et(Without lagged dependent variables on the RHS) Where 0 is known as the short run multiplier, or impact multiplier, because it gives the change in the mean value of Y following a unit change of X in the same time period. If the change of X is maintained at the same level thereafter, then, (0 + 1) gives the change in the mean value of Y in the next period, (0 + 1 + 2) in the following period, and so on. These partial sums are called interim, or intermediate, multiplier. Finally, after k periods, that is

i =0

k

i

= 0 + 1 + 2 + ... + k = B, therefore i is called the long run multiplier or

total multiplier, or distributed-lag multiplier. If define the standardized i* = i /i,, then it gives the proportion of the long run, or total, impact felt by a certain period of time. In order for the distributed lag model to make sense, the lag coefficients must tend to zero as k. This is not to say that 2 is smaller than 1; it only means that the impact of Xt-k on Y must eventually become small as k gets large. For example: a consumption function regression is written as Y = + 0.4Xt + 0.3 Xt-1 + 0.2 Xt-2 + 0.1Xt-3+ et Then the effect of a unit change of X at time t on Y and its subsequent time periods can be shown as the follow diagram: Effect on Y 0Xt 0 = 0.4 1 = 0.3 2 = 0.2 1Xt-1 2Xt-2 3Xt-3

Time t t-1 t-2 1 t-3

The multiplier round of the Investment injection $160 (k=1/(1-mpc) Period 1 2 3 4 5 6 7 1 => I $160 Y $160 80 40 20 10 5 1.25 160 C (if mpc = 0.5) $80 40 20 10 5 2.5

160

Other Examples: Money creation process, inflation process due to money supply, productivity growth due to expenditure or investment. Period 1 2 3 4 Total New Deposits $1000 900 810 729 $10,000 New Loans $900 810 729 656.1 $9,000 Required Reserve (10%) $100 90 81 72.9 $1,000

In general: Suppose the distributed-lag model is

Yt = + 0Xt +1Xt-1 +2Xt-2 ++ kXt-k + etThe basic idea: The long run responses of Y to a change in X are different from the immediate and short-run responses. And suppose the expect value in different periods is A. Then E(Xt) = E(Xt-1) = E(Xt-2) = A E(Yt) = + 0E(Xt) + 1E(Xt-1) + 2E(Xt-2) +.. + E(et) = + 0A + 1A + 2A + + 0 = + iA This gives the constant long run corresponding to X = A, and E(Y) will remain at this level unit when X changes again.

2

Suppose look for one period ahead (t+1), and Xt+1 = A+1 E(Yt+1) = + 0E(Xt+1) + 1E(Xt) + 2E(Xt-1) + + E(et+1) = + 0(A+1) + 1A +2A + + 0 = + 0 + 0A + 1A +2A + = + 0 + iA E(Yt+2) = + 0E(Xt+2) + 1E(Xt+1) + 2E(Xt) + +E(et+2) = + 0(A+1) + 1(A+1) +2A + + 0 = + 0 + 1 + 0A + 1A +2A + = + 0 + 1 + iA E(Yt+3) = + 0 + 1 + 2 + iA E(Yt+j) = + 0 + 1 + 2 + + j + iA For all periods after t+3, E(Y) remains unchanged at the level given by E(Yt+3), given no further change in X. Thus, E(Y) = + i X Reasons for Lags: Psychological reason: due to habit or inertia nature, people will not react fully to changing factors, e.g. Income, price level, money supply etc. Information reason: because imperfect information makes people hesitate on their full response to changing factors. Institutional reason: people cannot react to change because of contractual obligation. Ad Hoc Estimation of Distributed-Lag Models Estimation method: First regress Yt on Xt, then regress Yt on Xt and Xt-1, then regress Yt on Xt, Xt-1 and Xt-2, and so on. Y = + 0Xt Y = + 0Xt + 1Xt-1 Y = + 0Xt + 1 Xt-1 + 2 Xt-2 Y = + 0Xt + 1 Xt-1 + 2 Xt-2 + 3 Xt-3 Y = + 0Xt + 1 Xt-1 + 2 Xt-2 + 3 Xt-3 + This sequential procedure stops when the regression coefficients of the lagged variables start becoming statistically insignificant and / or the coefficient of at least one of the variables change signs, which deviate from our expectation. 3

Significant t-statistics of each coefficient The sign of i does not change Highest R2 and R 2 Min. values of BIC and/or AIC

Drawbacks: No a priori guide as to what is the maximum length of the lag. If many lags are included, then fewer degree of freedom left, and it makes statistical inference somewhat shaky. Successive lags may suffer from multi-collinearity, which lead to imprecise estimation. (When Cov(Xt-i, Xt-j) is high). Need long enough data to construct the distributed-lag model.

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I. Koyck Approach to Distributed-Lag Models (Geometrical lag Model):

Yt = + 0 Xt +0Xt-1 +02Xt-2 + + etor

Yt = + 0 0Xt +0 1Xt-1 +02Xt-2 + + etWhere i =0 and 0 < < 1 (A priori restriction on i by assumption that the i s follow a systematic pattern.) 0 1 2 3 4 5 6 7 0.8 0 0.8 0.64 0.512 0.409 0.327 0.262 0.209 . 0.3 0 0.3 0.09 0.027 0.0081 0.0024 0.0007 0.0002 i

k =0

k

= 0 (1 + + 2 + 3 + ...) = 0 (

1 ) which is the long run multiplier 1

Then

Yt-1 = + 0Xt-1 +0Xt-2 +02Xt-3 + + et-1 Yt-1 = + 0 Xt-1 + 02Xt-2 +03Xt-3 + + et-1and

(Yt -Yt-1) = ( - ) + + 0 Xt + (0Xt-1 - 0 Xt-1) + (02Xt-2 - 02Xt-2) + (03Xt-3 - 03Xt-3) ++(et -et-1)By Koyck transformation (from a distributed-lag model transformed into an autoregressive model):

Yt -Yt-1 =(1-) + 0Xt + (et-et-1) Yt = (1-) + 0Xt + Yt-1 + vt Yt = 0 + 1Xt + Yt-1 + vt The Yt-1 is a short run dynamic term and is built into the autoregressive model. The important of this autoregressive model gives the long-run multiplier that implied by the distributed lags model. The long-run multiplier can be obtained from the autoregressive model by calculating 1(1/(1-)). where vt ~ iid(0, 2)

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II. Rationalization of the Koyck Model: Adaptive Expectation Model

Yt =0 + 1X*t +et (Long run function, and X* is unobserved expected level)Go back one period and multiply (1-), it becomes:

(1-)Yt-1 =(1-)0 + 1(1-)X*t-1 + (1-)et-1 (to be subtracted later)By postulating:

X*t - X*t-1 = (Xt - X*t-1) X*t = Xt + (1-) X*t-1

where 0<