Outline. 11. Time Series Analysis. Basic Regression. Differences between Time Series and Cross Section

Transcription

1 Outline I. The Nature of Time Series Data 11. Time Series Analysis II. Examples of Time Series Models IV. Functional Form, Dummy Variables, and Index Basic Regression Numbers Read Wooldridge (2013), Chapter 10 2 I. The Nature of Time Series Data Differences between Time Series and Cross Section Time series A time series data set consists of observations on a variable or several variables over time. eg. GDP, money supply, interest rate. Cross section series A cross sectional data set is a data set collected from a population at a given point in time. (1) Ordering Cross section: Ordering usually is not important. Time series: A data set comes with a temporal ordering (2) Random sample Cross section: A random sample is drawn from the population. Each observation is randomly drawn (MLR.2). Time series: An observation is an outcome of random variables. I. The Nature of Time Series Data 3 I. The Nature of Time Series Data 4

2 Terms: II. Examples of Time Series Regression Stochastic random Realization observation Static and Finite Distributed Lag Models Formally, say observation A sequence of random variables indexed bime is called a stochastic process or a time series process. At a point in time, we obtain a single realization of the stochastic process. Note that we cannot go back in time. Static Model t = 1, 2,,n A static model is a model that relates o z using the same time period. 1 : An immediate effect of z on y. I. The Nature of Time Series Data 5 II. Examples of Time Series Regression 6 Examples of Time Series Regression Finite Distributed Lag (FDL) Models: Effect of one or more variables with a lag on y Example : Phillips Curve inf t unem t inf t : annual inflation rate (%) unem t : unemployment rate (%) Example: Effect of the growth in money supply on economic growth in Thailand over (quarterly data) ggdp: economic growth (percent) gm1: money supply growth (percent) = unem (s.e) (1.72) (.289) [t] [.828] {1.617} n=49 R 2 =.053 R 2 bar=.032 How to define 1? ggdp t gm1 t gm1 t 1 gm1 t 2 Interpret coefficients: let t = 0 0 : this quarter effect of MS growth on this quarter ggdp. 1 : last quarter effect of MS growth on this quarter ggdp 2 : two quarters ago effect of MS growth on this quarter ggdp. II. Examples of Time Series Regression 7 II. Examples of Time Series Regression 8

3 Example: effect of gm1 on ggdp 1 2 t = gm1 t +.121gm1 t 1.126gm1 t 2 p value {.88} {.0005} {.105} {.1359} n=39, R 2 =.5088 F=12.08 {p value= } Interpretation: Economic Significance. 1) Interpret 2) Interpret 3) Interpret A General form a FDL of order two 1 2 Interpret coefficients: 1) 0 : the immediate change in y due to the one unit increase in z at time t 0 is called the impact propensity or impact multiplier. 2) 0 : the long run change in y given a permanent increase in z 0 is called the long run propensity (LRP) or long run multiplier. II. Examples of Time Series Regression 9 II. Examples of Time Series Regression 10 A temporal increase in z: j 1 2 temporal means lasting only for a time. A permanent increase in z: Assumptions: 1) Before time t: z is a constant (c) 2) At time t, z increases one unit to c+1 3) After time t, z reverts back to its previous level, c Interpretation: 0 = 1 : immediate change in y due to the one unit increase in z at time t 1 = +1 1 : the change in y one period after the increase in z. 2 = +2 1 : the change in wo periods after the increase in z. Assumptions: 1) Before time t, z is a constant (c). 2) At time t, z increases one unit permanentlo c+1. Interpretation: With the permanent increase in z 0 = 1 : immediate change = +1 1 : increase in y after one period = +2 1 : increase in y after two periods is the LR change in y given a permanent increase in z in the FDL model of order 2. II. Examples of Time Series Regression 11 II. Examples of Time Series Regression 12

4 Example: effect of gm1 on ggdp Effect of Monetary Policy on Economic Growth 1 2 t = gm1 t +.121gm1 t 1.126gm1 t 2 p value {.88} {.0005} {.105} {.1359} n=39, R 2 =.5088 F=12.08 {p value= } Dependent Variable: GGDP Method: Least Squares Sample(adjusted): 1993:2 2002:4 Included observations: 39 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C Interpretation: Economic and Statistical Significance? Impact multiplier =? and Test? Long run multiplier =? and Test? GM GM1(-1) GM1(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Graph: a lag distribution with two nonzero lags. Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) II. Examples of Time Series Regression 13 II. Examples of Time Series Regression 14 under classical Assumptions Unbiasedness of OLS: TS.1 TS.3. TS.1: Linear in parameters. TS.2: No perfect collinearity No independent variable is constant or a perfect linear combination of the others. Given stochastic process {(x t1, x t2,,x tk, ): t = 1, 2,.., n} x t1 + + k x tk where {u t : t = 1, 2,., n} is the sequence of errors. x tj : t denotes the time period j indicates one of the k variables TS.3 : Zero Conditional Mean E(u t X) = 0 where X is an array with n rows and k columns

5 Example: =GDP; x t1 =M1 t x t2 =GE t (gov t spending); n=44, k=2 year = gdp x t1 = M1 x t2 = GE error 1960 gdp 60 M1 60 GE 60 u gdp 61 M1 61 GE 61 u gdp 62 M1 62 GE 62 u 62 Theorem 3.1: Unbiasedness of OLS If TS.1 TS.3 hold, then the OLS estimators are unbiased, conditional in X, i.e., gdp 01 M1 01 GE 01 u 01 E( ) = j 2002 gdp 02 M1 02 GE 02 u gdp 03 M1 03 GE 03 u 03 for j = 1,..., k u t is uncorrelated with each explanatory variable in every time period. We sahat x tj are strictly exogenous TS. 3' implies that u t is uncorrelated with regressors dated at time t, (TS.3' next chapter) E(u t x t1,,x tk ) = E(u t x) = 0 We say x tj are contemporaneously exogenous. year = gdp x t1 = M1 x t2 = GE error 2001 gdp 01 M1 01 GE 01 u 01 Why don t we assume E(u i X)=0 or strict exogeneity in the cross sectional analysis? Random sampling: u i is automatically independent of the explanatory variables other than i. Example: Strict Exogenity Assumption Model: ggdp t gm1 t TS. 3 requires that (1) u t and are uncorrelated (2) u t is also unrelated with past and future values of z; that is, z have no lagged effect on y. (3) A subtle point : the changes in error term today cannot cause future changes in z. (u t +1 ) (u t ggdp t gm1 t+1 ) This rules out feedback from y on future values of z

6 Example: Murder rate equation mrdrte t = polpc t mrdrte: murders per 10,000 people polpc: number of police in the force Two implications: (1) TS.3 implies that u t is uncorrelated with polpc in all time period. Violation: Higher u 0 may lead to larger polpc 1 force. Cases: Agricultural Production, rainfall and labor input Example : Agricultural Production (y) Case 1: y=output z=rainfall u t =error t=0 y 2000?? u 2000 (2) Explanatory variables that are strictly exogeneous cannot react to what has happened to y in the past. year=t =mrdrte t =polpc t u t =error ,000? 100? u Violation: high u 0 high mrdrt 0 high polpc 1 t=1 z 2001 Case 2: y=output z=labor input u t =error t=0 z 2000 u?? 2000 t=1 z TS.4: Homoskedasticity VAR(u t X) = VAR(u t ) = 2 t = 1,, n TS.5: No Serial Correlation Corr(u s,u t X) = 0 for all t s Example: Effect on Treasury bills i3 t inf t def t def t : federal deficit as a percentage of GDP. TS. 4 requires that unobservables affecting interest rates have a constant variance over time. Example: No Serial Correlation inf t def t When u t 1 > 0 then, on average, the error in the next period u t is positive, or Corr(u t, u t 1 ) > 0. This problem is called serial correlation or auto correlation. Violation: Variability of interest rates depends on the level of inflation or relative size of the deficits. This implies that if interest rate is high in this period, it will be high in the next period: a violation of TS

7 Serial Correlation Positive Serial Correlation Theorem 10.2: OLS Sampling Variances If TS.1 TS.5 hold, then Var( X ) = Negative Serial Correlation j = 1,, k, where SST j is the total sum of squares of x tj and R j2 is the regression of x j on the other regressors Theorem 10.3: Unbiased estimator of 2 If TS.1 TS.5 hold, then the unbiased estimator of 2 is 2 = Inference under the CLM Assumptions Assumption TS.6: Normality assumption The errors u t are independent of X and u t N(0, 2 ) Theorem 10.4: Gauss Markov Theorem If TS.1 TS.5 (Gauss Markov assumptions) hold, then conditional on X, the OLS estimators are the best linear unbiased estimators (BLUE). That is, u t is independently and identically distributed (i.i.d) as Normal (0, 2 )

8 Inference under the CLM Assumptions Theorem 10.5: Normal Sampling Distribution If TS.1 TS.6 (CLM assumptions) hold, then (1) is normally distributed, (2) t and F statistics have t and F distributions, respectively, under the null hypothesis, and (3) the usual construction of confidence interval is valid. Example: Effects of inflation and deficits on interest rate in the United States over i3: the three month T bill rate def: federal deficit as a percentage of GDP 3 t = inf t def t s.e. (0.44) (0.076) (0.118) t [2.84] [8.06] [5.93] n = 49 R 2 = R 2 bar= Interpret: Economic and Statistical Significance 1) Test inf and def at the 5% level. 2) Interpret the coefficient on inf 3) Interpret the coefficient on def IV. Functional Form, Dummy Variables, and Index number Index number: GDP Example Nominal GDP 1, , , ,615.4 Real GDP 1, , Index: GDP Deflator Change the base year NESDB uses 1988 as the base year for GNP deflator (base year) Suppose we want to change the base year to 1997 new index t old index old index t new base *100 Base period: 1988 Base value: (100/154)* (base year) (154/154)* (161/154)*100 IV. Functional Form, Dummy and Index 31 IV. Functional Form, Dummy and Index 32

9 Old and new GDP deflator Indexes compared. Calculate the growth rate of GDP deflator: another index for inflation Old GDP deflator Index New GDP deflator Index Old GDP deflator Index New GDP deflator Index Inflation rate Calculate the growth rate of GDP deflator: inflation new yeart old yeart growth rate ( )*100 old year t Inflation in 1999 = [( )/109.4]*100 = 4.3% IV. Functional Form, Dummy and Index 33 IV. Functional Form, Dummy and Index 34 Example : Effect of Personal Exemption on Fertility Rates; gfr t pe t ww2 t + 3 pill t Fertility Rate Equation with no lags Dependent Variable: GFR gfr t : general fertility rate the number of children born to every 1,000 women. pe t : real dollar value of the personal tax exemption ww2 = 1 during WW II ( ) = 0 otherwise ( , ) pill = 1 from 1963 on ( ) = 0 otherwise ( ) Method: Least Squares Sample: 1 72 Included observations: 72 Variable Coefficient Std. Error t-statistic Prob. C PE WW PILL Estimation: t = pe t 24.24ww2 t 31.59pill t s.e. (3.21) (.030) (7.46) (4.08) t {30.74} {2.77} {3.25} {7.74} n = 72 R 2 = R 2 bar = What can you say about statistical significance? IV. Functional Form, Dummy and Index 35 R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) 0 IV. Functional Form, Dummy and Index 36

10 Example: fertility rate equation with a lag Dependent Variable: GFR t = pe t.0058pe t pe t ww2 t 31.30pill t stat {29.07} {.579} {.030} {.269} {11.58} {7.86} n =70 R 2 = R 2 bar= Sample(adjusted): 3 72 Included observations: 70 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C PE Interpretation: 1) Interpret the coefficient on ww2. 2) Interpret the coefficient on pill. 3) Are pe t, pe t 1, pe t 2 individually statistically significant? 4) Are pe t, pe t 1, pe t 2 jointly statistically significant? Problem: multicollinearity?? PE(-1) PE(-2) WW PILL R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) 0 IV. Functional Form, Dummy and Index 37 IV. Functional Form, Dummy and Index 38 View/Coefficient Tests/Redundant Variables Redundant Variables: PE PE(-1) PE(-2) Long run propensity... gfr t pe t pe t 1 pe t 2 ww2 t + 3 pill t F-statistic Probability Log likelihood ratio Probability View/Coefficient Tests/Wald-Coefficient Restrictions t = pe t.0058pe t pe t ww2 t 31.30pill t {29.07} {.579} {.030} {.269} {11.58} {7.86} Wald Test: Equation: Untitled Test Statistic Value df Probability F-statistic (2, 64) Chi-square Null Hypothesis Summary: 5) Should we ditch pe t 1 or pe t 2? 6) What is the long run propensity in this model? 7) Is the long run propensity statistically significant? What is the null hypothesis? 8) Find a 95% confidence interval of the long run propensity. Normalized Restriction (= 0) Value Std. Err. C(3) C(4) Restrictions are linear in coefficients. IV. Functional Form, Dummy and Index 39 IV. Functional Form, Dummy and Index 40

11 Dependent Variable: GFR Sample(adjusted): 3 72 The 95% CI of LRP Included observations: 70 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C PE PE(-1)-PE PE(-2)-PE WW The 95% confidence interval of LRP is Given = and s.e.( ) = 0.030, the 95% CI is c*s.e.( ) =.101 2(.030) =.041 to.160 PILL R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic c: is the 95 th percentile in the t distribution with 64 DF n = 70 k = 5 n k 1 = = 64 c = 2.00 (DF = 60) = (DF = 90) Durbin-Watson stat Prob(F-statistic) 0 IV. Functional Form, Dummy and Index 41 IV. Functional Form, Dummy and Index 42 Recap of Time Series Analysis The Nature of Time Series Data Examples of Time Series Models Finite Sample Properties of OLS Functional Form, Dummy Variables, and Index Numbers 43