Reading Assignment Distributed Lag and Autoregressive Models ...
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Reading Assignment Distributed Lag and Autoregressive Models Chapter 17. Kennedy: Chapters 10 and 13.
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Distributed Lag and Autoregressive Models Distributed lag model: yt = + 0 xt + 1 xt-1 + ... + k xt-k + et . Autoregressive model: yt = + xt + yt-1 + et . Examine where the shock enters a resting model and measuring the impact on yt through time. These models work very well. Forecast well and explain well. But are hard to publish because our theory is fairly void of dynamics these empirical models dont necessarily come from economic models consistent with theory or similar empirical models can come from rather different economic models. There is also the following problem. There are fundamental conceptual differences, but similar empirical representations. This is a problem... Last, both sides of the model can be nonstationary that is not explained by a deterministic trend variable and then we can have a spurious regression.
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Interpretation of distributed lag model Impact Multipliers: Measure change in y given a change in x after: 0 period 0 1 period 1 ... k periods k k+1 periods 0 Total Impact: 0 + 1 + ... + k Interpretation of autoregressive model Cumulative Impact: Measure cumulative change in y given a change in x after: 0 period 1 period () 2 periods ( + 2) ... k periods ( + 2 + ... + k) Total Impact: i i = / (1 ). (And we could start the impacts, not in the 0th period, but in the 1st period.)
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Reasons for Distributed Lag and Other Dynamic Models 1. Psychological reasons: consumers form habits and producers must observe incentives repeatedly. 2. Technological reasons: technology may be slow to adopt or implement. 3. Institutional reasons: institutions may limit economic choices and speed of adjustment. Notice: these reasons are all ad hoc. Explicit dynamic models of economic behavior are needed. And are being developed. And are not empirically trivial...
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Start with a Distributed Lag Model yt = + 0 xt + 1 xt-1 + ... + k xt-k + et . How to choose k? Make use of institutional knowledge relevant to that market... But we must be careful about data mining, degrees of freedom, and collinearity. The following two lines are very, very, very important... Assume some structure on is. Specifically, k smaller than 0 (or 1 ), and the is transition back through time is smooth.
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Infinite Lag: Geometric Lag yt = + 0 xt + 1 xt-1 + 2 xt-2 + ... + et yt = + s xt-s + et s=0 assume s = 0 s where 0 < < 1 and s = 0, 1, 2,... yt = + 0 (xt + xt-1 + 2 xt-2 + ... ) + et yt = + 0 s xt-s + et s=0 (Count the parameters before and after)
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Transform to make estimable. yt = + 0 (xt + xt-1 + 2 xt-2 + ... ) + et (lag both sides) yt-1 = + 0 (xt-1 + xt-2 + ... ) + et-1 (Multiply by ) yt-1 = + 0 (xt-1 + 2 xt-2 +...) + et-1 Subtract 3rd equation from first and cancelling a lot of terms results in yt = (1 ) + 0 xt + yt-1 + (et et-1 ) which is okay in an undergraduate course to represent as yt = (1 ) + 0 xt + yt-1 + ut but not in a graduate course. Careful: error term contains a moving average component, lagged dependent variable is stochastic, and DW-d is invalid.
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Adaptive Expectations Economic Model: a rationalization for geometric lag. yt = + xt* + et Action by economic agent (yt ) depends on an unobservable expectations variable (xt* ) and assume xt* xt-1* = (xt xt-1* ) where 0 < 1 or xt* = xt + (1 ) xt-1* so that the expectations are a combination of actual conditions and previous expectations revising past expectations based on the current condition. Big Picture: We are starting with a structural model or an economic model and deriving a reduced form or an estimable econometric model. Then we will attempt to recover the structural parameters from the reduced form parameters.
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Substitute expectation equation into model yt = + [ xt + (1 ) xt-1* ] + et. Lag model one period, multiply lagged model by (1 ), and subtract result from model yt = + xt + (1 ) yt-1 + (et (1 )et-1 ). yt = 0 + 1 xt + 2 yt-1 + (et 2 et-1 ) (or yt = 0 + 1 xt + 2 yt-1 + ut but only for novices...) 2 = (1 ) so that = 1 2 1 = so that = 1 / So we say and are identified. This means that we can go from parameters estimated in the reduced form econometric model back to parameters in the structural economic model.
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Partial Adjustment Economic Model: Another rationalization. yt* = + xt + et Action by economic agent (yt* ) depends variable (xt ) but is only partial of what was intended yt - yt-1 = (yt* yt-1 ) where 0 < 1 or yt = yt* + (1 ) yt-1. Substitute model into adjustment equation yt = [ + xt + et ] + (1 )yt-1 yt = + xt + (1 )yt-1 + et . yt = 0 + 1 xt + 2 yt-1 + ut 2 = (1 ) so that = 1 2 1 = so that = 1 / and 0 = so that = 0 / .
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Finite Lag: Polynomial Distributed Lag This is a sharp contrast with the infinite lag approach. yt = + 0 xt + 1 xt-1 + ... + k xt-k + et k yt = + i xt-i + et i = 0 where i = 0 + 1 i + 2 i 2 +...+ m i m and m < k. Polynomial enforces a relationship between the is. Estimable model: assume an order of polynomial, substitute polynomial into model, solve for s. (Count the parameters before and after)
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Example: 3rd order polynomial and lag length of 5 5 yt = + i xt-i + et (Number of slope parameters?) i = 0 i = (0 + 1 i + 2 i 2 + 3 i 3) (Number of parameters with restriction?) 5 yt = + (0 + 1 i + 2 i 2 + 3 i 3) xt-i + et i = 0 5 5 5 5 yt = + 0 xt-i + 1 i xt-i + 2 i 2 xt-i + 3 i 3 xt-i + et i = 0 i = 0 i = 0 i = 0 yt = + 0 z0t + 1 z1t + 2 z2t + 3 z3t + et (Software often reports this model.) where z0t = xt + xt-1 + xt-2 + xt-3 + xt-4 + xt-5
z1t = xt-1 + 2 xt-2 + 3 xt-3 + 4 xt-4 + 5 xt-5 z2t = xt-1 + 4 xt-2 + 9 xt-3 + 16 xt-4 + 25 xt-5
z3t = xt-1 + 8 xt-2 + 27 xt-3 + 64 xt-4 + 125 xt-5
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Estimate s using Zs and recover s through restrictions i = 0 + 1 i + 2 i 2 + 3 i 3 0 = 0 1 = 0 + 1 + 2 + 3 2 = 0 + 1 2 + 2 4 + 3 8 3 = 0 + 1 3 + 2 9 + 3 27 4 = 0 + 1 4 + 2 16 + 3 64 k = ... V()s are recovered by using the formula for the variance of a random variable which is a linear combination of random variables. V(ki i ) = (SAS, EViews, and most packages report these also.) Computer software packages will perform polynomial distributed lagged regressions.
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Endpoint restrictions: i = 0 + 1 i + 2 i 2 + 3 i 3 Back: k+1 = 0 + 1 (k+1) + 2 (k+1) 2 + 3 (k+1) 3 = 0 Front: -1 = 0 - 1 + 2 - 3 = 0 Graphically,
-1 0 1 2 3 4 5 6 t Restrictions imply hypotheses which can be tested. Back has a lot of intuition but the front does not however, the mathematics works.
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Other Tests: Choosing order of m and lag length k? Polynomial order m: H0: m = 0 t-test. Lag order k: H0: k = 0 t-test. Be careful of data mining. Order m should be small but k could be large. Procedure: Choose a large lag length and polynomial order. (You must understand the market or action you are modeling. Study what you are modeling.) Test down starting with lag length and then test polynomial order. Stop where last lag and polynomial element are insignificant. Add endpoint restriction(s). Want polynomial and endpoints to be binding but not too binding not contradict the data. Do not want the kth variable or the mth polynomial element to be significant. Do not want to reject endpoint restrictions.
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Alternative Mechanical Procedure: Use an Information Criteria to determine lag length, Schwarz: SC = ln( 2 ) + k ln( T ) 2 maximum likelihood estimate of error variance k lag length T sample size. Choose k to minimize SC. Increasing k makes the model fit better but also makes the penalty go up. There are other Information Criteria. Many econometric packages will perform polynomial distributed lags and will test the order of the polynomial and endpoint restrictions. ex) EViews ls: y c pdl(x, Lag, Order, Restrictions) SAS has PROC PDLREG Practically, Need good reason for polynomial greater than 3rd order. Use back endpoint restrictions.
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Autoregressive Models Geometric lag: yt = (1 ) + 0 xt + yt-1 + (et et-1 ) Adaptive expectations: yt = + xt + (1 ) yt-1 + (et (1 ) et-1 ) Partial adjustment: yt = + xt + (1 ) yt-1 + et Autoregressive model: yt = + xt + yt-1 + ut A lot of dynamic models look alike in empirical implementation. Especially, if the error term has serial correlation. So, its tough to recover the structural economic model from the econometric time series model. Thus, your alternative models are not different enough for statistical methods to say which is correct...
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Problems with using OLS with autoregressive model: by definition we may make the error term: ut = et et-1 Serial correlation: E(ut ut-1 ) = - 2