Introduction to Modern Time Series Analysis

Transcription

1 Introduction to Modern Time Series Analysis Gebhard Kirchgässner, Jürgen Wolters and Uwe Hassler Second Edition Springer 3 Teaching Material The following figures and tables are from the above book. They are provided to help instructors and students and may be used for teaching purposes as long as a reference to the book is given in class. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

2 Introduction and Basics bn Euro year Figure.: Real Gross Domestic Product of the Federal Republic of Germany in billions of Euro, 9 bn Euro year Figure.: Quarterly Changes of the Real Gross Domestic Product (ΔGDP) of the Federal Republic of Germany, 9 99 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

3 percent Figure.3: Quarterly Growth Rates of the Real Gross Domestic Product (qgr) of the Federal Republic of Germany, 9 99 year bn Euro year Figure.: Annual Changes of the Real Gross Domestic Product (Δ GDP) of the Federal Republic of Germany, 9 99 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

4 percent Figure.5: Annual Growth Rates of Real Gross Domestic Product (agr) of the Federal Republic of Germany, 9 99 bn Euro Figure.: Smooth Component and actual values of the Real Gross Domestic Product of the Federal Republic of Germany, 9 year year Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

5 Figure.7: Example of a Random Walk where only the steps + and are possible Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

6 CHF/USD percent a) Exchange Rate CHF/USD 97 year year - -5 ˆ b) Continuous Returns CHF/USD 5 5 c) Estimated Autocorrelations Figure.: Exchange Rate Swiss Franc U.S. Dollar, Monthly data, January 97 to December Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

7 Univariate Stationary Processes 7.5 x t 5.5 t a) Realisation b) Theoretical autocorrelation function ˆ c) Estimated autocorrelation function with confidence intervals Figure.: AR() process with α =.9 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

8 x t t - - a) Realisation b) Theoretical autocorrelation function ˆ c) Estimated autocorrelation function with confidence intervals Figure.: AR() process with α=.5 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

9 x t 5.5 t a) Realisation 5 5 b) Theoretical autocorrelation function ˆ c) Estimated autocorrelation function with confidence intervals Figure.3: AR() process with α = -.9 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

10 Percent year ˆ ( ) ˆ ( ) a) Popularity of the CDU/CSU, b) Autocorrelation ( ) and partial ( ) autocorrelation functions with confidence intervals 5 5 c) Estimated autocorrelation function of the residuals of the estimated AR()-process with confidence intervals Figure.: Popularity of the CDU/CSU, 97 9 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

11 x t 5 t -5 - a) Realisation b) Theoretical autocorrelation function ˆ c) Estimated autocorrelation function with confidence intervals Figure.5: AR() process with α =.5, α = -.5 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

12 x t 5.5 t a) Realisation b) Theoretical autocorrelation function ˆ c) Estimated autocorrelation function with confidence intervals Figure.: AR() process with α =. and α = -.5 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

13 Percent year ˆ ( ) ˆ ( ) a) Three month money market rate in Frankfurt b) Estimated autocorrelation ( ) and partial autocorrelation ( ) functions with confidence intervals 5 5 c) Estimated autocorrelation function of the residuals of the estimated AR()-process with confidence intervals Figure.7: Three month money market rate in Frankfurt, Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

14 kk kk kk kk k 5 5 AR() process with =.9 k 5 5 AR() process with = -.9 k 5 5 AR() process with =.5, = -.5 k 5 5 AR() process with =., = -.5 Figure.: Estimated partial autocorrelation functions Table.: Characteristics of the Autocorrelation and the Partial Autocorrelation Functions of AR and MA Processes Autocorrelation Function Partial Autocorrelation Function MA(q) breaks off with q does not break off AR(p) does not break off breaks off with p Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

15 Figure.9: Theoretical autocorrelation functions of ARMA(,) processes Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

16 Percent ˆ a) New York three month money market rate, b) Autocorrelation ( ) and partial ( ) autocorrelation functions of the first differences with confidence intervals 5 5 c) Autocorrelation function of the residuals of the estimated ARMA(,)-process with confidence intervals year Figure.: Three month money market rate in New York, 99 3 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

17 Table.: Forecasts of the Council of Economic Experts and of the Economic Research Institutes Period R RMSE MAE ME â U *.5*.57 Institutes Council of Economic Experts * *.7*.7* * *.5*.77* -.73*.35.55* *.5*.5*.*.3*.* * denotes the better of the two forecasts. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

18 3 Granger Causality percent Growth Rate of Real GDP - - Interest Rate Spread (t-) (GLR - GSR) year Figure 3.: Growth rate of real GDP and the four quarters lagged interest rate spread in the Federal Republic of Germany, (in percent) Table 3. Test for Granger Causality (I): Direct Granger Procedure /5 /9, Observations y x k k F(y x) F(y x) F(y x) ln(gdp r ) ln(m r ).7*** **.3. ln(gdp r ) GLR GSR 3.* 3.35**..97(*).77*.79 ln(m r ) GLR GSR 5.5***.9.99**.5*.7 5.5*** (*), *, **, or *** denote that the null hypothesis that no causal relation exists can be rejected at the, 5, or. percent significance level, respectively. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

19 Figure 3.. (The dotted lines are the approximate 95 percent confidence intervals.) ˆρ(k) k Figure 3.a: Cross-correlations between the residuals of the univariate models of GDP and the quantity of money M ˆρ(k) k Figure 3.b: Cross-correlations between the residuals of the univariate models of GDP and the interest rate spread Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

20 ˆρ(k) k Figure 3.c: Cross-correlations between the residuals of the univariate models of the quantity of money M and the interest rate differential Table 3.: Test for Granger Causality (II): Haugh-Pierce Test /5 /9, Observations Y x ˆ () k S(y x) S(y x) S(y<=>x) ln(gdp r ) ln(m r ).79(*).57** ** 7.3*.5 3.* ln(gdp r ) GLR GSR.7.3.*.(*).7 3.7(*) 5.55(*) ln(m r ) GLR GSR.33***.97* 9.* 3.95***.(*).7.3** (*), *, **, or. *** denote that the null hypothesis that no causal relation exists can be rejected at the, 5, or. percent significance level, respectively. Table 3.3: Optimal Lag Length for the Hsiao Procedure Akaike Criterion Schwarz Criterion Relation * k * * k k * k * * k k ln(m r ) ln(gdp r ) ln(gdp r ) ln(m r ) 5 3 (GLR GSR) ln(gdp r ) ln(gdp r ) (GLR GSR) Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

21 Table 3.: Models Estimated with the Hsiao Procedure /5 /9, Observations Criterion Akaike Criterion Schwarz Criterion Explanatory Variable Dependent Variable ln(gdp r,t ) ln(m r,t ) ln(gdp r,t ) ln(m r,t ) Constant term ln(gdp r, t- ) ln(gdp r,t- ) ln(gdp r,t-3 ) ln(m r,t- ) ln(m r,t- ) ln(m r,t-3 ) ln(m r,t- ) ln(m r,t-5 ) ln(m r,t- ) ln(m r,t-7 ) ln(m r,t- ). (.7).75*** (3.59).59*** (.).3*** (3.) -.95 (.3) -.3 (.5).39* (.5).7*** (.73) -.73 (.9).5 (.3) -.7*** (3.53).3* (.5) -. (.3).9 (.) -.3* (.).3 (.).75*** (3.).59*** (.) ˆ (û,û ) *** (3.9).97*** (.) -.35 (.99).3 (.) -.5** (.7) R SE Q(m) 3.*.* 3.3*.5* m The numbers in parentheses are the absolute values of the estimated t statistics. (*), *, **, or *** denote that the corresponding null hypothesis can be rejected at the, 5, or. percent significance level, respectively. m denotes the number of degrees of freedom of the Q statistic. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

22 Table 3.5: Models Estimated with the Hsiao Procedure /5 /9, Observations Criterion Akaike Criterion Schwarz Criterion Explanatory Variable Dependent Variable ln(gdp r,t ) (GLR GSR) t ln(gdp r,t ) (GLR GSR) t Constant term ln(gdp r, t- ) ln(gdp r,t- ) ln(gdp r,t-3 ).37 (.7).73*** (.).** (.) -.3 (.5) -.3* (.). (.3) ln(gdp r,t- ).5* (.5) ln(gdp r,t-5 ) -.3(*) (.7) (GLR GSR) t- -.5 (.) (GLR GSR) t-.** (.).*** (.9) -. (.7) (GLR GSR) t ** (.9) (GLR GSR) t-.*** (3.7) (GLR GSR) t-5 -.7** (.95).3 (.3).733*** (.7) -.3 (.3).3* (.) ˆ (û,û ) ** (.93).3*** (.3) -.9 (.) -.3* (.3).** (3.5) -.37*** (3.53) R SE Q(m) m 7 7 The numbers in parentheses are the absolute values of the estimated t statistics. (*), *, **, or *** denote that the corresponding null hypothesis can be rejected at the, 5, or. percent significance level, respectively. m denotes the number of degrees of freedom of the Q statistic. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

23 Table 3.: Test for Granger Causality: Direct Granger Procedure with Three Variables /5 /9, Observations Y x z k F(y x) F(y x) F(y x) ln(gdp r ) ln(mr) GLR GSR.77* 3.7**.577.**.3*.7 ln(gdp r ) GLR GSR ln(m r )..(*).7.3.7(*).9 ln(m r ) GLR GSR ln(gdp r ) 7.5*** *** 3.3**.9.5*** (*), *, **, or *** denote that the null hypothesis that no causal relation exists can be rejected at the, 5, or. percent significance level, respectively. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

24 Vector Autoregressive Processes. Response of X to X. Response of X to X Response of X to X Response of X to X Figure.: Impulse response functions Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

25 Accumulated Response of X to X Accumulated Response of X to X Accumulated Response of X to X Accumulated Response of X to X Figure.: Cumulative impulse response functions Response of DLGDPR to DLGDPR. Response of DLGDPR to DLMR.. Response of DLGDPR to GLSR Response of DLMR to DLGDPR 3 3 Response of DLMR to DLMR 3 Response of DLMR to GLSR Response of GLSR to DLGDPR. Response of GLSR to DLMR. Response of GLSR to GLSR Figure.3: Impulse response functions Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

26 Accummulated Response of DLGDPR to DLGDPR Accummulated Response of DLGDPR to DLMR Accummulated Response of DLGDPR to GLSR Accummulated Response of DLMR to DLGDPR Accummulated Response of DLMR to DLMR Accummulated Response of DLMR to GLSR Accummulated Response of GLSR to DLGDPR - - Accummulated Response of GLSR to DLMR - - Accummulated Response of GLSR to GLSR Figure.: Cumulative impulse response functions Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

27 Table.: Variance Decomposition Forecast horizon x x Immediate periods periods periods Infinity x.. x x x x x x x 9.3. x 5..9 x Table.a: Variance Decomposition /5 /9, Observations Forecast horizon Δ ln(gdp r ) Δ ln(m r ) GLR GSR Δ ln(gdp r ) immediate Δ ln(m r ) GLR GSR... Δ ln(gdp r ) year Δ ln(m r ) GLR GSR Δ ln(gdp r ) years Δ ln(m r ) GLR GSR Δ ln(gdp r ) years Δ ln(m r ) GLR GSR Δ ln(gdp r ) infinity Δ ln(m r ) GLR GSR Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

28 Table.b: Variance Decomposition /5 /9, Observations Forecast horizon Δ ln(gdp r ) Δ ln(m r ) GLR GSR Δ ln(gdp r ) immediate Δ ln(m r )... GLR GSR Δ ln(gdp r ) year Δ ln(m r ) GLR GSR Δ ln(gdp r ) years Δ ln(m r ) GLR GSR Δ ln(gdp r ) years Δ ln(m r ) GLR GSR Δ ln(gdp r ) infinity Δ ln(m r ) GLR GSR Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

29 5 Nonstationary Processes Figure 5.: Linear and quadratic trend, superimposed by a pure random process Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

30 Figure 5.: Realisations of AR() processes α =.3 (------), α=.97 ( ) Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

31 Figure 5.3: Random walk with (-----) and without ( ) drift Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

32 First differences of the random walk First differences of the model with a linear trend Original residuals Original residuals Figure 5.: Scatter diagrams of the first differences against the original residuals of nonstationary processes Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

33 Residuals of the model with a stochastic trend Residuals of the model with a time trend Original residuals Original residuals Figure 5.5: Scatter diagrams of the residuals of regressions on a time trend against the original residuals of nonstationary processes Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

34 Actual and estimated values - - Residuals Model with a linear trend 5 Actual and estimated values Residuals Model with a random walk with drift Figure 5. Actual and estimated values and residuals of the models with linear deterministic and stochastic trends Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

35 Constant term linear trend Table 5.: Results of Linear Trend Elimination ( Observations) Model with a linear trend random walk random walk with drift.993 (99.) 5.7 (9.79) 9.73 (.9).9 (9.55) 3 (.3) ).7.9 (59. R Durbin-Watson Density of the estimated coefficient compared with a normal distribution with the same variance and = Density of the t statistic Figure 5.7: Density of the estimated autocorrelation coefficient and the t statistic under the null hypothesis of a random walk. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

36 UER GER/EER SER Figure 5.: Development of the Swiss, German/European and US Euromarket interest rates. Monthly data, January 93 December Table 5.: Results of the Augmented Dickey-Fuller Tests /93 /, Observations Variable Levels. Differences k Test Statistic k Test Statistic SER (.7) -7. (.) GER/EER (.7) -.9 (.) UER (.755) -. (.) The tests were performed for levels with as well as for first differences without a constant term. The numbers in parentheses are the p values. The number of lags, k, has been determined with the Hannan-Quinn criterion. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

37 Density of the estimated coefficient compared with a normal distribution with the same variance and μ = Density of the Dickey-Fuller t statistic Figure 5.9a: Density of the estimated coefficient and of the t statistic for the null hypothesis of an AR() process with ρ =.95 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

38 Density of the estimated coefficient compared with a normal distribution with the same variance and μ = Density of the Dickey-Fuller t statistic Figure 5.9b: Density of the estimated coefficient and of the t statistic for the null hypothesis of an AR() process with ρ =.9 Table 5.3: Results of Unit Root and Stationarity Tests for Inflation /99 9/99, 5 Observations m/k United States United Kingdom France Germany Italy Phillips- Perron -.95** -.** -9.3** -.5** -5.** -.** -.3** -.5** -.** -7.39** KPSS.**.5*.**.5**.57**.9**.**.**.9**.5** ADF 3 -.3** -3.* -.** -.97* -.7(*) ** -3.9** -3.3* (*),,* or,** denote that the corresponding null hypothesis can be rejected at the, 5, or percent significance level, respectively. Source: U. HASSLER and J. WOLTERS (995, Tables 3 and, p. 39). Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

39 Figure 5.a: German Inflation Rate: Actual values ( ), permanent component according to S. BEVERIDGE and CH.R. NELSON ( ), permanent component according to R.J. HODRICK and E.C. PRESCOTT ( - - ) Figure 5.b: German Inflation Rate: cyclical component according to S. BEVERIDGE and CH.R. NELSON ( ), cyclical component according to R.J. HODRICK and E.C. PRESCOTT ( ) Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

40 Figure 5.a: Swiss real money balances M, 9 : Actual values ( ) and permanent component ( ) due to the Hodrick-Prescott filter Figure 5.b: Swiss real money balances M, 9 : Actual values ( ) and permanent component ( ) due to the Hodrick-Prescott filter Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

41 Cointegration Density of the t statistic.... Density of the R Density of the Durbin-Watson statistic Figure.: Densities of the estimated t value, R, and the Durbin-Watson statistic Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

42 Table.: Critical Values of the Dickey-Fuller Test on Cointegration in the Static Model α k 3 Model with constant term Model with constant term and time trend The values for k = are the critical values of the Dickey-Fuller unit root test. Source: J.G. MACKINNON (99, Table, p. 75). Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

43 ln(m ) t year a) Logarithm of the per capita real quantity of money M ln(y ) t percent b) Logarithm of the per capita real GNP year c) Long-run interest rate year Figure.: Data for the Federal Republic of Germany, 9 99 Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

44 Table.: Critical Values of the Cointegration Test in the Error Correction Model α k 3 Model with constant term Model with constant term and time trend Source: A. BANERJEE, J.J. DOLADO and R. MESTRE (99, Table, pp. 7f.) percent year Figure.3: German three month money market rate in Frankfurt Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

45 Table.3: Results of the Johansen Cointegration Test Model Hypotheses Eigenvalues Trace Test max Test r (.).75 (.) z, z 3 r..3 (.97).3 (.97) r (.) 33. (.) z, z r.9.7 (.9).7 (.9) The numbers in parentheses are the p values of the corresponding statistics. Table.: Results of the Johansen Cointegration Test Model Hypotheses Eigenvalues Trace Test max Test r..57 (.) 5.5 (.) z, z 3, z VECM() r (.) 9.3 (.) r.. (.9). (.9) r (.) 9.7 (.) z, z 3, z VECM() r (.) (.) r..73 (.93).73 (.93) The numbers in parentheses are the p values of the corresponding statistics. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

46 7 Nonstationary Panel Data Australia Canada Denmark Germany Japan New Zealand Norway Sweden Switzerland United Kingdom United States Figure 7.: year government bond yields, January 99 December Table 7.: p values of the Augmented Dickey-Fuller Tests for year government bonds p p p Australia.53 Japan.35 Switzerland.5 Canada.3 New Zealand. United Kingdom.39 Denmark.339 Norway.77 United States.9 Germany.5 Sweden. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

47 Figure 7.: Cross-sectional average of the U.S. spreads Table 7.: CADF Test statistics for spreads against the U.S. t ρ k i t ρ k i Australia -.9 New Zealand -.55 Canada -3.9 Norway -3.3 Denmark -3.9 Sweden -.7 Germany -5.5 Switzerland -3. Japan -. United Kingdom -.9 Table 7.3: Ordered p values for Augmented Dickey-Fuller tests for the spreads with α =. p (j) j α/ p (j) j α/ New Zealand.9. Norway.55. Switzerland.. Denmark.7.7 Australia..3 Sweden.37. United Kingdom.37. Japan.35.9 Germany.35.5 Canada.77. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

48 Table 7.: Ordered p values for tests for no cointegration ˆ j p (j) ˆ j p (j) Denmark.5.7 Switzerland.9.7 Sweden.95. Australia.5.75 United Kingdom.5. Canada New Zealand.. Norway Germany.3. Japan.. Table 7.5: Estimates of the error correction adjustment coefficient i OLS SUR OLS SUR i Australia -.5 (-.77) -.7 (-5.) New Zealand -. (-.75) -. (-3.3) Canada -.3 (-.5) -. (-3.) Norway -.5 (-.97) -. (-.) Denmark -.5 (-3.7) -.79 (-.5) Sweden -.3 (-3.3) -.3 (-.9) Germany -.3 (-.) -.3 (-5.7) Switzerland -. (-.3) -.75 (-.7) Japan -. (-.3) -.37 (-.5) United Kingdom -. (-.9) -.57 (-.3) The numbers in parentheses are the t statistics of the estimated parameters. Table 7.: Estimates of the short-run US influence b i * b i b i * b i Australia. (.39).9 (.3) New Zealand.75 (.9). (.3) Canada. (.9).5 (.73) Norway.3 (.).9 (.5) Denmark.7 (.9).3 (.) Sweden. (.).9 (.57) Germany.7 (.3).7 (3.) Switzerland.7 (.93). (.9) Japan.3 (.).3 (.) United Kingdom.9 (.).7 (.7) The numbers in parentheses are the t statistics of the estimated parameters. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

49 Table 7.7: SUR error correction estimates i,ec i,ec Australia. New Zealand.5 Canada.957 Norway.3 Denmark.33 Sweden.3 Germany.9 Switzerland.93 Japan.59 United Kingdom. Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

50 Autoregressive Conditional Heteroscedasticity 7 July July 997 October 997 October 99 January 99 9 May 999 a) German Stock Market Index: Data January 99 9 May 999 b) German Stock Market Index: Continuous returns c) German Stock Market Index: Histogram of the continuous returns Figure.: German Stock Market Index, January 99 until 9 May 999, observations Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3

51 . ˆ( ).. ˆ( ) d) Estimated autocorrelations of the residuals... e) Estimated autocorrelations of the squared residuals Figure.: German Stock Market Index, January 99 until 9 May 999, observations (continued).5 Modified t-distribution..3.. Standard normal distribution Figure.:Density functions of a normalised t distribution with 5 degrees of freedom, variance one and a standard normal distribution Introduction to Modern Time Series Analysis, Springer-Verlag Berlin Heidelberg 3