Reading Assignment. Distributed Lag and Autoregressive Models. Chapter 17. Kennedy: Chapters 10 and 13. AREC-ECON 535 Lec G 1

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1 Reading Assignment Distributed Lag and Autoregressive Models Chapter 17. Kennedy: Chapters 10 and 13. AREC-ECON 535 Lec G 1

2 Distributed Lag and Autoregressive Models Distributed lag model: y t = α + β 0 x t + β 1 x t β k x t-k + e t. Autoregressive model: y t = α + x t β + φ y t-1 + e t. Examine where the shock enters a resting model and measuring the impact on y t 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 don t 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. AREC-ECON 535 Lec G 2

3 Interpretation of distributed lag model Impact Multipliers: Measure change in y given a change in x after: 0 period β 0 1 period β 1... k periods β k k+1 periods 0 Total Impact: β 0 + β β k Interpretation of autoregressive model Cumulative Impact: Measure cumulative change in y given a change in x after: 0 period β 1 period β (φ) 2 periods β (φ + φ 2 )... k periods β (φ + φ φ k ) Total Impact: β i φ i = β / (1 φ). (And we could start the impacts, not in the 0 th period, but in the 1 st period.) AREC-ECON 535 Lec G 3

4 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... AREC-ECON 535 Lec G 4

5 Start with a Distributed Lag Model y t = α + β 0 x t + β 1 x t β k x t-k + e t. 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 β i s. Specifically, β k smaller than β 0 (or β 1 ), and the β i s transition back through time is smooth. AREC-ECON 535 Lec G 5

6 Infinite Lag: Geometric Lag y t = α + β 0 x t + β 1 x t-1 + β 2 x t e t y t = α + β s x t-s + e t s=0 assume β s = β 0 ω s where 0 < ω < 1 and s = 0, 1, 2,... y t = α + β 0 (x t + ω x t-1 + ω 2 x t ) + e t y t = α + β 0 ω s x t-s + e t s=0 (Count the parameters before and after ) AREC-ECON 535 Lec G 6

7 Transform to make estimable. y t = α + β 0 (x t + ω x t-1 + ω 2 x t ) + e t (lag both sides) y t-1 = α + β 0 (x t-1 + ω x t ) + e t-1 (Multiply by ω) ωy t-1 = ωα + β 0 (ωx t-1 + ω 2 x t ) + ωe t-1 Subtract 3 rd equation from first and cancelling a lot of terms results in y t = α (1 ω) + β 0 x t + ω y t-1 + (e t ωe t-1 ) which is okay in an undergraduate course to represent as y t = α (1 ω) + β 0 x t + ω y t-1 + u t but not in a graduate course. Careful: error term contains a moving average component, lagged dependent variable is stochastic, and DW-d is invalid. AREC-ECON 535 Lec G 7

8 Adaptive Expectations Economic Model: a rationalization for geometric lag. y t = α + β x t + e t Action by economic agent (y t ) depends on an unobservable expectations variable (x t ) and assume x t x t-1 = φ (x t x t-1 ) where 0 < φ 1 or x t = φ x t + (1 φ ) x t-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. AREC-ECON 535 Lec G 8

9 Substitute expectation equation into model y t = α + β [φ x t + (1 φ ) x t-1 ] + e t. Lag model one period, multiply lagged model by (1 φ), and subtract result from model y t = φα + φβ x t + (1 φ ) y t-1 + (e t (1 φ)e t-1 ). y t = β 0 + β 1 x t + β 2 y t-1 + (e t β 2 e t-1 ) (or y t = β 0 + β 1 x t + β 2 y t-1 + u t but only for novices...) β 2 = (1 φ) so that φ = 1 β 2 β 1 = φβ so that β = β 1 / φ So we say φ and β 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. AREC-ECON 535 Lec G 9

10 Partial Adjustment Economic Model: Another rationalization. y t = α + β x t + e t Action by economic agent (y t ) depends variable (x t ) but is only partial of what was intended y t - y t-1 = δ (y t y t-1 ) where 0 < δ 1 or y t = δ y t + (1 δ ) y t-1. Substitute model into adjustment equation y t = δ [α + β x t + e t ] + (1 δ)y t-1 y t = δ α + δ β x t + (1 δ )y t-1 + δ e t. y t = β 0 + β 1 x t + β 2 y t-1 + u t β 2 = (1 δ) so that δ = 1 β 2 β 1 = δ β so that β = β 1 / δ and β 0 = δ α so that α = β 0 / δ. AREC-ECON 535 Lec G 10

11 Finite Lag: Polynomial Distributed Lag This is a sharp contrast with the infinite lag approach. y t = α + β 0 x t + β 1 x t β k x t-k + e t k y t = α + β i x t-i + e t i = 0 where β i = ω 0 + ω 1 i + ω 2 i ω m i m and m < k. Polynomial enforces a relationship between the β i s. Estimable model: assume an order of polynomial, substitute polynomial into model, solve for ω s. (Count the parameters before and after ) AREC-ECON 535 Lec G 11

12 Example: 3rd order polynomial and lag length of 5 5 y t = α + β i x t-i + e t i = 0 (Number of slope parameters?) β i = (ω 0 + ω 1 i + ω 2 i 2 + ω 3 i 3 ) (Number of parameters with restriction?) 5 y t = α + (ω 0 + ω 1 i + ω 2 i 2 + ω 3 i 3 ) x t-i + e t i = y t = α + ω 0 x t-i + ω 1 i x t-i + ω 2 i 2 x t-i + ω 3 i 3 x t-i + e t i = 0 i = 0 i = 0 i = 0 y t = α + ω 0 z 0t + ω 1 z 1t + ω 2 z 2t + ω 3 z 3t + e t (Software often reports this model.) where z 0t = x t + x t-1 + x t-2 + x t-3 + x t-4 + x t-5 z 1t = x t x t x t x t x t-5 z 2t = x t x t x t x t x t-5 z 3t = x t x t x t x t x t-5 AREC-ECON 535 Lec G 12

13 Estimate ω s using Z s and recover β s through restrictions β i = ω 0 + ω 1 i + ω 2 i 2 + ω 3 i 3 β 0 = ω 0 β 1 = ω 0 + ω 1 + ω 2 + ω 3 β 2 = ω 0 + ω ω ω 3 8 β 3 = ω 0 + ω ω ω 3 27 β 4 = ω 0 + ω ω ω 3 64 β k =... V(β)s are recovered by using the formula for the variance of a random variable which is a linear combination of random variables. V( k i ω i ) = (SAS, EViews, and most packages report these also.) Computer software packages will perform polynomial distributed lagged regressions. AREC-ECON 535 Lec G 13

14 Endpoint restrictions: β i = ω 0 + ω 1 i + ω 2 i 2 + ω 3 i 3 Back: β k+1 = ω 0 + ω 1 (k+1) + ω 2 (k+1) 2 + ω 3 (k+1) 3 = 0 Front: β -1 = ω 0 - ω 1 + ω 2 - ω 3 = 0 Graphically, β t Restrictions imply hypotheses which can be tested. Back has a lot of intuition but the front does not however, the mathematics works. AREC-ECON 535 Lec G 14

15 Other Tests: Choosing order of m and lag length k? Polynomial order m: H 0 : ω m = 0 t-test. Lag order k: H 0 : β 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. AREC-ECON 535 Lec G 15

16 Alternative Mechanical Procedure: Use an Information Criteria to determine lag length, Schwarz: SC = ln( σ 2 ) + k ln( T ) σ 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. AREC-ECON 535 Lec G 16

17 Autoregressive Models Geometric lag: y t = α (1 ω) + β 0 x t + ω y t-1 + (e t ω e t-1 ) Adaptive expectations: y t = φα + φβ x t + (1 φ ) y t-1 + (e t (1 φ ) e t-1 ) Partial adjustment: y t = δ α + δ β x t + (1 δ ) y t-1 + δ e t Autoregressive model: y t = α + x t β + φ y t-1 + u t A lot of dynamic models look alike in empirical implementation. Especially, if the error term has serial correlation. So, it s 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... AREC-ECON 535 Lec G 17

18 Problems with using OLS with autoregressive model: by definition we may make the error term: u t = e t λ e t-1 Serial correlation: E(u t u t-1 ) = -λ σ 2 Errors correlated with independent variable: Cov(y t-1, u t ) = -λ σ 2 biased and inconsistent estimates. Serial correlation in a model with a lagged dependent variable is very very very bad. AREC-ECON 535 Lec G 18

19 Tests for Serial Correlation 1. Durbin h statistic: h T ( utu t1) ˆ where ˆ and... y 2 t1 1 T ( Var( ˆ)) ( u ) t... h ~ N(0,1) under H 0 : ρ = 0. Notice the test is invalid if Var(φ) > BG test. AREC-ECON 535 Lec G 19

20 What to do if you have serial correlation in autoregressive model? First, theory better say that is the model. Second, use an instrumental variable. Need a variable correlated with y t-1 but not with u t. y t = f(y t-1, z 1t, z 2t,... ) + u t y t = g(x t, x t-1,... ) + v t More on instrumental variables with simultaneous equations... Third, maximum likelihood. AREC-ECON 535 Lec G 20

21 Spatial Models Spatial Autocorrelation y = X β + ε and ε = λwε + v or y i = x i β + ε i ε i = λ Σ j w ji ε ji + v i Spatial Autoregression y = X β + ρwy + ε y i = x i β + ρw ji y ji + ε i where W is a weighting matrix that determines neighbors and possibly distance. Suppose you have cross sectional data (e.g., U.S. counties or states). Then could you pick up one and place it somewhere else in the country and get the same result? Doubtful. That s spatial dependence. AREC-ECON 535 Lec G 21

22 What s W look like with temporal data? Temporal neighbors are easy. W What s W look like with temporal data? Spatial neighbors less so. You are going to need software. W AREC-ECON 535 Lec G 22