11.1 Gujarati(2003): Chapter 12
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1 11.1 Gujarati(2003): Chapter 12
2 Time Series Data 11.2 Time series process of economic variables e.g., GDP, M1, interest rate, echange rate, imports, eports, inflation rate, etc. Realization An observed time series data set generated from a time series process Remark: Your age is not a realization of time series process.
3 11.3
4 Decomposition of time series 11.4 X t Trend Cyclical or seasonal random time
5 Static Models 11.5 Y t = β 1 + β 2 Z t + u t Subscript t indicates time. The regression is a contemporaneous relationship, i.e., how does current Y affected by current Z? Eample: Static Phillips curve model inf t = β 1 + β 2 unem t + u t inf: inflation rate unem: unemployment rate
6 Finite Distributed Lag effects 11.6 Effect at time t Effect at time t+1 Effect at time t+2. Effect at time t+q Economic action at time t The Distributed Lag Effect (with order q) Y t =α 1 +δ 2 Z t +u t Y t+1 =α 1 +δ 2 Z t+1 +δ 3 Z t +u t or Y t =α 1 +δ 2 Z t +δ 3 Z t-1 + u t. Y t+q =α 1 +δ 2 Z t+q +..+δ q Z t +u t or Y t =α 1 +δ 2 Z t + +δ q Z t-q +u t
7 11.7 Economic action at time t The Distributed Lag Effect Effect at time t-1 Effect at time t-2 Effect at time t-3. Effect at time t-q Y t = α 1 +δ 2 Z t +δ 3 Z t-1 +δ 4 Z t-2 + +δ q Z t-q +u t Initial state: z t = z t-1 = z t-2 = c
8 11.8 Y = α 1 + δ 2 Z t + δ 3 Z t-1 + δ 4 Z t-2 + u t Long-run propensity (LRP) = δ 2 + δ 3 + δ 4 Permanent change in Y for 1 unit permanent change in Z. Distributed Lag model in general: Y t = α 1 + δ 2 Z t + δ 3 Z t δ q Z t-q + other factors + u t LRP (or long run multiplier) = δ 2 + δ δ q
9 Time Trends 11.9 Linear time trend Y t = α 1 + α 2 t + u t Constant absolute change Eponential time trend ln(y t ) = α 1 + α 2 t + u t Constant growth rate Quadratic time trend Y t = α 1 + α 2 t + α 3 t 2 + u t Accelerate change
10 Definition: First-order of Autocorrelation, AR(1) Y t = β 1 + β 2 X 2t + u t t = 1,,T If Cov (u t, u s ) = E (u t u s ) 0 and if u t = ρ u t-1 + v t where -1 < ρ < 1 (ρ : RHO) and v t ~ iid (0, σ v2 ) (white noise) where t s E ( v This scheme is called first-order autocorrelation and denotes as AR(1) Autoregressive : The regression of u t can be eplained by itself lagged one period. ρ(rho) : the first-order autocorrelation coefficient or coefficient of autocovariance var( cov( t ) v t v t = ), v 0 = σ s ) 2 v = 0
11 Eample of serial correlation: Year Consumption t = β 1 + β 2 Income t + error t u u u u u u u Error term represents other factors that affect consumption TaRate 1999 TaRate 2000 The current year Ta Rate may be determined by previous year rate TaRate 2000 = ρ TaRate v 2000 u t = ρ u t-1 + v t ν t ~ iid(0, σ v2 )
12 Why does serial correlation occur? Inertia. A salient feature of most economic time series is inertia, or sluggishness. As is well known, time series such as GNP, price indees, production, employment, and unemployment ehibit (business) cycles. Specification Bias: Ecluded Variables Case. This is the case of ecluded variable specification bias. Often the inclusion of such variables removes the correlation pattern observed among the residuals. For eample, suppose we have the following demand model: Specification Bias: Incorrect Functional Form. Suppose the true or correct model in a cost-output study is as follows:
13 Why does serial correlation occur? v i will have autocorrelation
14 Why does serial correlation occur? Cobweb Phenomenon. The supply of many agricultural commodities reflects the so-called cobweb phenomenon, where supply reacts to price with a lag of one time period because supply decisions take time to implement (the gestation period). Lags. In a time series regression of consumption ependiture on income, it is not uncommon to find that the consumption ependiture in the current period depends, among other things, on the consumption ependiture of the previous period. That is,
15 Why does serial correlation occur? Manipulation of Data. In empirical analysis, the raw data are often manipulated. Eample: Converting monthly data to quarterly introduces smoothness into the data by dampening the fluctuations in the monthly data. Therefore, the graph plotting the quarterly data looks much smoother than the monthly data, and this smoothness may itself lend to a systematic pattern in the disturbances, thereby introducing autocorrelation. Data Transformation. it can be shown that the error term v t is autocorrelated.
16 Why does serial correlation occur? Nonstationarity. a time series is stationary if its characteristics (e.g., mean, variance, and covariance) are time invariant; that is, they do not change over time. If that is not the case, we have a nonstationary time series.
17 If u t = ρ 1 u t-1 + v t it is AR(1), first-order autoregressive If u t = ρ 1 u t-1 + ρ 2 u t-2 + v t it is AR(2), second-order autoregressive. High order autocorrelation If u t = ρ 1 u t-1 + ρ 2 u t ρ p u t-p + v t it is AR(p), p th -order autoregressive Autocorrelation AR(1) : Cov (u t u t-1 ) > 0 => 0 < ρ < 1 Cov (u t u t-1 ) < 0 => -1 < ρ < 0 positive AR(1) negative AR(1) -1 < ρ < 1
18 0 ^u i ^u i Positive autocorrelation time 0 ^u i Cyclical: Positive autocorrelation Positive autocorrelation time 0 time The current error term tends to have the same sign as the previous one.
19 ^u i Negative autocorrelation time The current error term tends to have the opposite sign from the previous. ^u i No autocorrelation 0 time The current er. term tends to be randomly moving wrt to the previous term
20 Positive vs. Negative Autocorrelation ρ > 0 ρ < 0
21 The meaning of ρ : The error term u t at time t is a linear combination of the current and past disturbance < ρ < 1-1 < ρ < 0 ρ = 1 The further the period is in the past, the smaller is the weight of that error term (u t-1 ) in determining u t, past is less important than current The past is as equally important as the current. ρ > 1 The past is more importance than the current.
22 The consequences of serial correlation: The estimated coefficients are still unbiased. E(β ^ k ) = β k ^ 2. The variances of the β k is no longer the smallest 3. The standard error of the estimated coefficient, se(β ^ k ) becomes large Therefore, when the regression has AR(1) errors, The estimators are not BLUE.
23 Two variable regression model: Y t = β 1 + β 2 X 2t + u t The OLS estimator of β 2, ^ Σ y ===> β 2 = Σ If E (u t u t-1 ) = 0 then Var (β ^ σ 2 ) = 2 Σ 2 t If E(u t u t-1 ) 0, and u t = ρu t-1 + v t, then var (β ^ σ 2 ) AR1 = 2 2σ + 2 Σ ρ t t+1 Σ + ρ 2 t t+2 Σ 2 t Σ 2 t Σ 2 t Σ 2 t < ρ < 1 If ρ = 0, zero autocorrelation, than Var(β ^ 2 ) AR1 = Var(β ^ 2 ) If ρ 0, autocorrelation, than Var(β ^ 2 ) AR1 Var(β^ 2 ) The AR(1) variance is not correct and may not be the smallest
24 Autoregressive scheme: u t = ρu t-1 + v t ==> u t = ρ[ρu t-2 + v t-1 ] + v t ==> u t-1 = ρu t-2 + v t-1 u t = ρ 2 u t-2 + ρv t-1 + v t ==> u t-2 = ρu t-3 + v t-2 => u t = ρ 2 [ρu t-3 + v t-2 ] + ρv t-1 + v t u t = ρ 3 u t-3 + ρ 2 v t-2 + ρv t-1 + v t σ 2 =var(u t )=E(u t u t ) = σ v ρ 2 E(u t u t-1 ) = ρσ 2 E(u t u t-2 ) = ρ 2 σ 2. E(u t u t-s ) = ρ s σ 2 It means the more we go to past, the less effect on current period
25 Gujarati(2003) Table12.4, pp.460 How to detect autocorrelation? 11.25
26 Run OLS: u ˆ ρuˆ + v and check the t-value of the coefficient t = t 1 t DW = ˆ ρ 1 = 1 =
27 Durbin-Watson Autocorrelation test From OLS regression result: where d or DW * = Check DW Statistic Table (At 5% level of significance, k =k-1=1, n=40) d L = d u = Reject H 0 region H 0 : no autocorrelation ρ = 0 H 1 : yes, autocorrelation eists. or ρ > 0 positive autocorrelation d L d u DW *
28 Durbin-Watson test OLS : Y = β 1 + β 2 X β k X k + u t obtain ^u t, DW-statistic(d) -1 < ρ < 1 Assuming AR(1) process: u t = ρu t-1 + v t I. H 0 : ρ = 0 no autocorrelation H 1 : ρ > 0 yes, positive autocorrelation DW * Compare d * and d L, d u (critical values) if d * < d L ==> reject H 0 if d * > d u ==> not reject H 0 if d L d * d u ==> this test is inconclusive
29 Durbin-Watson test(cont.) T (u ^ ^ t u t-1 ) 2 t=2 DW = 2 (1 -^ρ) T ^ u 2 (d) t t=1 d = 2 (1- ρ) ^ d ==> = 1 - ρ^ Since -1 ρ ^ 1 implies 0 d 4 ==> ρ ^ = 1- d 2
30 Durbin-Watson test(cont.) II. H 0 : ρ =0 no negative autocorrelation H 1 : ρ < 0 yes, negative autocorrelation we use (4-d) (when d is greater than 2) if (4 - d) < d L or 4 - d L < d ==> reject H 0 if (4 - d) > d u or 4 - d u > d ==> not reject H 0 if d L (4 - d) d u or 4 - d u d 4 - d L ==> inconclusive
31 Durbin-Watson test(cont.) II. H 0 : ρ =0 No autocorrelation H 1 : ρ 0 two-tailed test for auto correlation either positive or negative AR(1) If d < d L or d > 4 - d L ==> reject H 0 If d u < d < 4 - d u ==> not reject H 0 If d L d d u ==> inconclusive or 4 - d u d 4 - d L
32 H 0 : ρ = 0 positive autocorrelation H 1 : ρ > 0 H 0 : ρ = 0 negative autocorrelation H 1 : ρ < reject H 0 not reject not reject reject H 0 inconclusive inconclusive 0 d L d u d u 4-d L DW (d) 1% & 5% Critical values
33 For eample : UM ^ t = CAP t CAP t T t (15.6) (2.0) (3.7) (10.3) _ R 2 = 0.78 F = 78.9 σ^ u = RSS = 29.3 DW = 0.23 n = 68 (i) K = 3 (number of independent variable) (ii) n = 68, α= 0.01 significance level 0.05 (iii) d L = 1.525, d u = d L = 1.372, d u = Reject H 0, positive autocorrelation eists
34 The assumptions underlying the d(dw) statistics : 1. Intercept term must be included in OLS regression. 2. X s are nonstochastic 3. Only test AR(1) : u t = ρu t-1 + v t where v t ~ iid (0, σ v2 ) Not include the lagged dependent variable, Y t = β 1 + β 2 X t2 + β 3 X t3 + + β k X tk + γ Y t-1 + u t (autoregressive model) 5. No missing observation missing N.A. N.A. 82 N.A. N.A Y X......
35 Breusch-Godfrey (BG) test of higher-order autocorrelation or called Durbin s m test (Lagrange Multiplier, LM, Test) Test Procedures: (1) Run OLS and obtain the residuals u^ t. (2) Run ^u t against all the regressors in the model plus the additional regressors, u^ t-1, u^ t-2, u^ t-3,, u^ t-p. u^ t = β 1 + β 2 X t + u^ t-1 + u^ t-2 + u^ t u^ t-p + v Obtain the R 2 value from this regression. (3) compute the BG-statistic: (n-p)r 2 (4) compare the BG-statistic to the χ 2 p (p is # of degree-order) (5) If BG > χ 2 p, reject Ho (No autocorrelation), it means there is a higher-order autocorrelation If BG < χ 2 p, not reject Ho, it means there is a no higher-order autocorrelation
36 11.36 Click VIEW
37 11.37 Compare with the critical values Check on the t-statistics To see The order of autocorrelation
38 Remedy: 1. First-difference transformation Y t = β 1 + β 2 X t + u t Y t-1 = β 1 + β 2 X t-1 + u t-1 assume ρ = 1 ==> Y t - Y t-1 = β 1 - β 1 + β 2 (X t - X t-1 ) + (u t - u t-1 ) ==> Y t = β 2 X t + u t no intercept Y t = β 1 + β 2 X t + β 3 T + u t 2. Add T = trend Y t-1 = β 1 + β 2 X t-1 + β 3 (T -1) + ε t-1 ==> (Y t - Y t-1 ) = (β 1 - β 1 ) + β 2 (X t - X t-1 ) + β 3 [T - (T -1)] + (u t - u t-1 ) ==> Y t = β 2 X t + β 3 *1 + u t ==> Y t = β 1* + β 2 X t + u t If ^β 1* > 0 => an upward trend in Y ^ (β 2 > 0)
39 3. Cochrane-Orcutt Two-step procedure (CORC) (1). Run OLS on and obtains ^u t Y t = β 1 + β 2 X t + u t (2). Run OLS on ^u t = ρ u^ t-1 + v t and obtains ρ^ (3). Use the ^ρ to transform the variables : Generalized Least Squares (GLS) method Y t* = Y t - ρ^ Y t-1 Y t = β 1 + β 2 X t + u t X t* = X t - ^ -) ^ρ Y ρ X t-1 = β 1 ρ ^ + β 2 ρ^ X t-1 + ρu ^ t-1 t-1 (Y t - ρy ^ t-1 )= β 1 (1-ρ) ^ +β 2 (X t - ρx ^ t-1 ) + (u t -ρu ^ t-1 ) (4). Run OLS on Y t* = β 1* + β 2* X t* + u t *
40 4. Cochrane-Orcutt Iterative Procedure (5). If DW test shows that the autocorrelation still eisting, than it needs to iterate the procedures from (4). Obtain the u * t = Y t - β 1* - β 2* X t (6). Run OLS ^ u t* = ρ ^u t-1* + v t ^ DW 2 ρ (1 - ) 2 and obtain ρ^ which is the second-round estimated ρ (7). Use the ^ρ to transform the variable ^ Y ** t = Y t - ρ Y t-1 Y t = β 1 + β 2 X t + u t X ** t = X t ^ - ρ X t-1 ^ ρ Y t-1 = β 1 ρ ^+ β 2 ρx ^ ^ t-1 + ρu t-1 (8). Run OLS on Y ** t = β ** 1 + β ** 2 X ** t + u ** t Where is ^ ^ ^ (Y ^ t - ρ Y t-1 ) = β 1 (1 - ρ) + β 2 (X t - ρ X t-1 ) + (u t - ρ u t-1 ) 11.40
41 Cochrane-Orcutt Iterative procedure(cont.) (9). Check on the DW 3 -statistic, if the autocorrelation is still eist, than go into the third-round and so on. ^ ^ Until (ρ - ρ < 0.01), the estimated ρ s differ a little Generalized least Squares (GLS) 5. Prais-Winsten transformation Y t = β 1 + β 2 X t + u t t = 1,,T (1) Assume AR(1) : u t = ρu t-1 + v t -1 < ρ < 1 ρy t-1 = ρβ 1 + ρβ 2 X t-1 + ρu t-1 (2) (1) - (2) => (Y t - ρy t-1 ) = β 1 (1 - ρ) + β 2 (X t - ρx t-1 ) + (u t - ρu t-1 ) GLS => Y t* = β 1* + β 2* X t* + u* t
42 To avoid the loss of the first observation of each variable, the first observation of Y * and X * should be transformed as : Y t=1* = 1 - ρ^ 2 (Y t=1 ) X t=1* = 1 - ρ^ 2 (X t=1 ) but Y t=2* = Y t=2 - ^ρ Y t=1 ; X t=2* = X t=2 - ^ρ X t=1 Y t=3* = Y t=3 - ^ρ Y t=2 ; X t=3* = X t=3 - ^ρ X t= Y * t = Y t - ^ρ Y t-1 ; X * t = X t - ρ^ X t-1
43 6. Durbin s Two-step method : Y t = β 1 + β 2 X t + u t Since (Y t - ρy t-1 ) = β 1 (1 - ρ) + β 2 (X t - ρx t-1 ) + V t => Y t = β 1* + β 2 X t - ρβ 2 X t-1 + ρy t-1 + V t I. Run OLS => this specification Y t = β 1* + β 2* X t - β 3* X t-1 + β 4* Y t-1 + v t Obtain II. Transforming the variables : III. Run OLS on model : β ^ 4* as an estimated ^ρ (RHO) Y t* = Y t - β ^ 4* Y t-1 as Y t* = Y t - ρ ^ Y t-1 ^ and X t* = X t - β 4* X t-1 as X t* = X t - ρ^ X t-1 Y t* = α 1 + α 2 X t* + u t where α^ 1 = β^ 1 (1 - ρ) and α^ 2 = β^ 2 Compare
44 Lagged Dependent Variable and Autocorrelation Y t = β 1 + β 2 X 2 t + β 3 X 3 t + + β k X k.t + α 1 Y t-1 +u t DW statistic will often be close to 2 or DW does not converge to 2 (1 -^ρ) Durbin-h Test: Compute h * = ^ρ DW is not reliable n 1 - n*var (α^ 1 ) Compare h * to Z where Z c ~ N (0,1) normal distribution If h * > Z c => reject H 0 : ρ = 0 (no autocorrelation)
45 Eample: Gujarati(2003) Table 12-4, p Wage(Y t ) = β 1 + β 2 Output(X t ) + u t DW 2(1 ^- ρ) ( )
46 11.46 u t = ρ u t-1 + v t ^ ρ 1 - DW ( )
47 Cochrane-Orcutt Two-step procedure (2) Critical values: D u =1.337 D L =1.237 Since DW > D u No Autocorrelation After CO-correction
48 Running the Cochrane-Orcutt iterative procedure in EVIEWS 11.48
49 11.49 ρ^^
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