Introduction to Time Series Analysis of Macroeconomic- and Financial-Data. Lecture 5: Trends, Model Selection, and Summary

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1 Introduction Introduction to Time Series Analysis of Macroeconomic- and Financial-Data Felix Pretis Programme for Economic Modelling Oxford Martin School, University of Oxford Lecture 5: Trends, Model Selection, and Summary Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

2 Re-Cap So far: Auto-regressive Models Forecasting works well if nothing changes over time...(i.e. world is stationary) But...things change all the time! (think back 10, 20, 100 years) World is non-stationary!...how to model? Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

3 Today AR(1) model has three elements: (1) Where Y t was the last time period. (2) The unexpected event ɛ t. (3) Constant term allowing mean of Y t to be non-zero. Y t = β }{{} 1 + β 2 Y }{{ t 1 } (3) (1) + ɛ t }{{} (2), ɛ t N0, σ 2. (1) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

4 Do things remain the same, or change through time? We expect Y t to change through time (hence ɛ t ). But we expect it to revert to its equilibrium over time: Equilibrium value = mean value. But what about structural change? Financial crises? Does economic progress mean that equilibria change? Stationarity: Underlying distribution of Y t is time invariant. All aspects of distribution: Mean, variance, skewness, kurtosis. In practice we focus on Mean and variance. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

5 A Stationary Series Always has the same data generating process: No structural change taking place. Model as: Y t = β 1 + β 2 Y t 1 + ɛ t, iid(0, σ 2 ). (2) Stationarity implies that β 2 < 1: Shocks die away. Stationarity implies that: E(Y t ) = µ y = β 1 1 β 2. (3) Long-run equilibrium that economic variable returns to. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

6 Stationary Series Forecasts DLCPI US Inflation Without shocks we expect series to settle down to equilibrium. For US inflation that mean is (EY t ) = ˆβ 0 /(1 ˆβ 1 ) = Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

7 Forms of Non-Stationarity Stationary is the exception not the norm! Types of Non-stationarity: like going to zoo to look at non-elephants 1... Many many forms... = Three common forms encountered and tame-able: 1: Linear Trends 2: Structural Breaks (sudden change in parameters) 3: Unit-roots/Stochastic trends (Advanced) 1 with thanks to Anders Rahbek Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

8 1: Linear Trends 750 Wheat Price Index UK Example: Price of Wheat in the UK from Series trending downwards - need to account for trend Can include a linear trend as regressor Y t = β 1 + β 2 t + ɛ t (4) Can treat as normal regressor (standard inference applies) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

9 Model Fit: Trend vs. Constant Wheat Fitted r:wheat (scaled) Wheat Fitted 2 r:wheat (scaled) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

10 Spurious Correlation What happens if we use non-stationary trending variables in our regression models? Y t = β 1 + β 2 X t + ɛ t (5) Suppose β 2 = 0 (no relationship between Y t and X t : if: Y t and X t are stationary (stable) no problem if Y t is trending and X t is trending risk of spurious relationship! Find more spurious correlations: Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

11 Spurious Relationship: Example Japanese GDP and Cumulative Rainfall in Fortaleza (NE Brazil) Japan RGDP 6000 Cumulative Rainfall Fortaleza Brazil Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

12 Model RGDP using Rainfall Japan RGDP Model Fit Residuals Model Jap.RDGP t = (4938) (1.45) cumrainfall Brazil t R 2 =0.98, F(1,36) = 2206 [0.000]** Cumulative Rainfall highly significant, explains 98% of variation in Japanese GDP! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

13 Taming the trend Need to account for the trend - two options: Model the trend: include a trend variable Remove the trend: difference the data, use Y t, X t DJapan GDP DCumulativeRainfall_Brazil Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

14 Option 1: Model the Trend Jap.RGDP = 9467 (8497) + 11 (28.9) cumrainfall Brazil t (4679) Trend t ˆβ 2 = 11 with (se = 28.9) not statistically significant anymore! The estimation sample is: Coefficient Std.Error t-value t-prob Constant cumrainfall_brazil Trend Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

15 Option 2: Remove the Trend by Differencing By taking differences we can remove the trend: Subtract Y t 1 from both sides: Y t = β 1 + β 2 X t + β 3 t + ɛ t (6) Y t Y t 1 = β 1 + β 2 X t + β 3 t + ɛ t Y t 1 This is equal to: Y t = β 1 + β 2 X t + β 3 t + ɛ t (β 1 + β 2 X t 1 + β 3 (t 1) + ɛ t 1 ) and simplifies to: The trend is gone! Y t = β 2 X t + β 3 + ɛ t 1 Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

16 Option 2: Differencing Japanese RGDP and Brazilian Rainfall Japan RGDP 5000 Cumulative Rainfall D Japan RGDP D Cumulative Rainfall Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

17 Option 2: Differencing Model in differences: Jap.RGDP = (2976) (18.3) cumrainfall Brazil t Coefficient Std.Error t-value t-prob Constant DcumRainfall_Brazil D Japan RGDP Fitted No significant relationship! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

18 Model Building How to build a model: 1 Come up with an interesting question (hard!) 2 Collect the data 3 Plot the data! relationship and functional forms Over time: trending? stable? Scatter plots: relationship? linear? non-linear? Time series properties: PACF 4 Build general model estimate Diagnostic tests satisfied? Can the model explain the data? Yes? Hooray! (very very rare...) No? Modify model and repeat. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

19 Automatic Model Selection Build general model estimate modify: labour intensive! Automatic Model Selection Start with very general model ( General Unrestricted Model ) Drop variables through path search General to Specific Making sure diagnostic tests satisfied Finds simplest possible econometric model that is well specified Autometrics in OxMetrics Note: Does not replace you: garbage in, garbage out! Reduces hard work selecting correct model. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

20 Autometrics in Practice 1 Start with general model: All possibly important variables and lags (can fix by setting as U - fixed ). 2 At next menu select Autometrics box; choose p-value. Drops variables until all significant at chosen p α (e.g. 0.05) 3 Click OK and OK and wait for OxMetrics to select final model. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

21 What does Autometrics do? Tree-Search drop variables and make sure models well specified. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

22 Autometrics General Model Many lags, many potentially important variables Search Algorithm Estimates many models starting from the general model Tests model diagnostics Tests variables for significance Final Model Smallest, well-specified model Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

23 Automatic Model Selection So far: Japanese Kuznets curve model mis-specified: Coefficient Std.Error t-value t-prob Lco2_pc_ Constant Lrgdp_pc Lrgdp_pc_sq AR 1-2 test: F(2,44) = [0.0310]* ARCH 1-1 test: F(1,48) = [0.3590] Normality test: Chiˆ2(2) = [0.9730] Hetero test: F(5,44) = [0.0762] Hetero-X test: F(8,41) = [0.0128]* RESET23 test: F(2,44) = [0.0050]** How can we fix it? let the data speak: automatic model selection. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

24 Automatic Model Selection Start with general model for log(co2): log(co2) t 1 (for auto-correlation) log(rgdp) t, log(rgdp) t 1 log(rgdp) 2 t, log(rgdp)2 t 1 log(rgdp) t, log(rgdp) t 1 (economic growth?) General Model: log(co2) t = β 1 + β 2 log(co2) t 1 + β 3 log(gdp) t +β 4 log(gdp) t 1 + β 5 log(gdp) 2 t +β 6 log(gdp) 2 t 1 + β 7 log(gdp) t +β 8 log(gdp) t 1 + ɛ t using Autometrics... Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

25 Automatic Model Selection Autometrics Selection at p = 0.01 (1%): General Model: log(co2) t = β 1 + β 2 log(co2) t 1 + β 3 log(gdp) t +β 4 log(gdp) t 1 + β 5 log(gdp) 2 t +β 6 log(gdp) 2 t 1 + β 7 log(gdp) t +β 8 log(gdp) t 1 + ɛ t Estimated 14 different models until reduced to Specific Model: log(co2) t = β 1 + β 2 log(co2) t 1 + β 7 log(gdp) t + ɛ t What is left? Persistence and Economic growth! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

26 Automatic Model Selection Coefficient Std.Error t-value t-prob Lco2_pc_ DLrgdp_pc Constant U AR 1-2 test: F(2,44) = [0.1448] ARCH 1-1 test: F(1,47) = [0.3569] Normality test: Chiˆ2(2) = [0.2452] Hetero test: F(4,44) = [0.4854] Hetero-X test: F(5,43) = [0.6360] RESET23 test: F(2,44) = [0.1415] log(co2) Fitted scaled residual Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

27 Specific Model log(co2) = (0.0219) T=49 R 2 =0.987 log(co2) t (0.206) AR 1-2 test: F(2,44) = [0.1448] ARCH 1-1 test: F(1,47) = [0.3569] Normality test: Chiˆ2(2) = [0.2452] Hetero test: F(4,44) = [0.4854] Hetero-X test: F(5,43) = [0.6360] RESET23 test: F(2,44) = [0.1415] Maybe no Kuznets curve? High persistence: ˆβ 1 =0.99 Economic Growth Effect: log(rgdp) t, ˆβ 7 =1.21 increased CO2 emissions during boom decreased emission during recession log(rgdp) t (0.0495) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

28 CO2 Emissions and Economic Growth kg CO2 Real GDP CO2 Real GDP Real GDP 0.1 D log(real GDP) log(usd) 0.0 Recession Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

29 Autometrics has more uses Automatic Model Selection to detect which variables matter to find well-specified models to detect when changes/structural breaks happen? Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

30 Structural Breaks Parameters not constant: β β e.g. Temporary shift in mean Already encountered two potential examples: Japanese Exports 0.5 D12LJapanExports Fitted Residuals Baseball Attendance %Attendance Fitted Residuals (scaled) Large Residual! Large Residual! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

31 Correcting for Breaks Dummy Variable: = 1 during break = 0 otherwise accounts for all variation of these observations = as if removed from the sample and model estimated outside of the break period E.g. Dummy: 2008:6 2011: D12LJapanExports Dummy: 2008: : Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

32 Detection of Breaks How can we detect and control for breaks? Presence of breaks affects parameter estimates Parameter estimates affect whether we can detect a break simultaneity problem, must do estimation and detection jointly! Impulse Indicator Saturation: add indicator (0/1) variable for every observation Keep only significant ones easy to do with Autometrics (Automatic Model Selection Algorithm in OxMetrics/PcGive) Choose: Outlier and Break Detection IIS Searches over full set of dummy variables keeps significant Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

33 Japanese Exports Model: T 12 Y t = α 12 Y t 1 + µ + 1 i=t δ i i=1 Estimated Model & Detected Breaks (at 0.1%): D12LJapanExports = 0.9 (0.019) (0.049) (0.0026) D12LJapanExports t (0.049) I:2009(11) t (0.049) I:2008(11) t 0.22 (0.049) I:2009(12) t (0.049) I:2010(1) t I:2009(1) t identifies break period from: 2008: (1) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

34 Japenese Exports 0.5 D12LJapanExports Fitted r:d12ljapanexports (scaled) I:2008(11) I:2009(1) I:2009(11) I:2009(12) I:2010(1) Detected Breaks: Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

35 Baseball Attendance %Attendance Fitted Residuals (scaled) Large Residual! Large Residual! Breaks? Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

36 Baseball Attendance %Attend. t = α 1 %Attend. t 1 + β 1 + β 2 PCT t + β 3 Unemp. t + ɛ t %Attend. = (0.0913) ( ) %Attend. t ( ) Unempl. t ( ) PCT t AR 1-2 test: F(2,45) = [0.5510] ARCH 1-1 test: F(1,49) = [0.4158] Normality test: Chiˆ2(2) = [0.0036]** Hetero test: F(6,44) = [0.7810] Hetero-X test: F(9,41) = [0.9109] RESET23 test: F(2,45) = [0.8994] What now?? Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

37 Baseball: Break detection Use Impulse Indicator Saturation: %Attend. t = α 1 %Attend. t 1 +β 1 +β 2 PCT t +β 3 Unemp. t + Detected two outliers or breaks: in 1967 and 1981! %Attend. t = 0.65 (0.087) 0.48 (0.16) (0.31) α 1 %Attend. t (0.097) (0.28) PCT t 1 PCT t (0.0083) T 1 i=t δ i +ɛ t i=1 I:1967 t 0.42 (0.09) Unemp. t I:1981 t Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

38 Baseball Attendance Breaks detected: 1967 (+0.33), 1981 (-0.42) what happened? 1967: Red Sox reach World Series for first time in 20 years 1981: Players Strike (38% Games cancelled!) Corrected for Breaks - Effects: Coefficient on unemployment significant! ˆβ 3 = 0.017, se=0.0083, = p ˆβ 3 = Diagnostic Tests AR 1-2 test: F(2,42) = [0.7260] ARCH 1-1 test: F(1,49) = [0.3001] Normality test: Chiˆ2(2) = [0.7246] Hetero test: F(8,40) = [0.7220] Hetero-X test: F(14,34) = [0.1216] RESET23 test: F(2,42) = [0.1907] Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

39 Structural Breaks Sudden changes/shocks are common! Need to be controlled for! Can be detected (IIS, many other ways!) Correct parameter estimates Interpretation of shock itself! (e.g. baseball, recession...) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

40 Practical: Automatic Selection Automatic Model Selection - Kuznets Curve Use Autometrics at 1% General Model: Constant (U-fixed) log(co2) t 1 log(rgdp) t, log(rgdp) t 1 log(rgdp) 2 t, log(rgdp)2 t 1 log(rgdp) t, log(rgdp) t 1 Does model selection ever result in an exact Kuznets curve specification? Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

41 Practical: First Differences Model in first differences Estimate the Kuznets curve in first differences! Interpret significance and diagnostics! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

42 Practical: Structural Breaks Dealing with structural breaks: Constructing Dummy Variables: common approach if known time period is problematic (e.g. World War II) Detection of Breaks: finding breaks/disturbances without prior knowledge of their occurrence Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

43 Practical: Baseball Model Load data red sox 1948.in7 Estimate the ADL model: %Attend. t = α 1 %Attend. t 1 + β 1 + β 2 PCT t + β 3 Unemp. t + ɛ t Recall two drastic events: 1967: World Series for first time! 1981: Player s Strike Construct 2 dummy variables using calculator: I:1967 I:1981 Estimate the model with dummies: %Attend. t = α 1 %Attend. t 1 + β 1 + β 2 PCT t + β 3 Unemp. t +δ 1 I : δ 2 I : ɛ t Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

44 Practial: Fitted Model 1.00 att_pop100 Fitted Note the perfect fit, observations "dummied" out! r:att_pop100 (scaled) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

45 How could we have found these breaks? Using break detection methods: e.g. IIS (Impulse Indicator Saturation) Re-estimate the above model using IIS Mark all regressors as U - fixed. Select IIS in Automatic Model Selection Choose p α = 0.01 Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

46 Practial: Additional Useful Features: Useful features of OxMetrics (not covered so far): Aggregate: change frequencies of measurement Batch files: store model code in file to be replicable (ALT+B) Test - Further Output: write model results in equation format Ox Programming Language: more complex models Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

47 Re-Cap of the Course Themes: OLS Regression Model Mis-specification Dependence over time Forecasting Non-Stationarity Cointegration Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

48 OLS Regression Assumptions: (i) (Y t, X t, Z t ) independent across t. (ii) Identical conditional distribution: (Y t X t, Z t ) (β 1 + β 2 X t + β 3 Z t, σ 2 ). (iii) X t and Z t exogenously determined for Y t. (iv) A parameter space exists: β 1, β 2, β 3, σ 2 R R +, Gives model: Y t = β 1 + β 2 X t + β 3 Z t + ɛ t, ɛ t N0, σ 2 (7) Parameter interpretation as before except ceteris paribus: EY t X t, Z t X t = β 2, (8) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

49 Misspecification Conclusions drawn from model only valid if model is well-specified! 1 Heteroskedasticity (variance changes through sample) (Hetero) 2 Errors normally distributed (Normality) 3 Functional form of the model (RESET) 4 Residual Auto-correlation (AR) 5 Variance persistent (ARCH) How to spot: Plot the series, fitted values, and residuals! OxMetrics reports mis-specification tests by default! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

50 Depdendence over Time Time series = dependence over time between observations. Has to be modelled! How to spot dependence and determine lag length: (Partial) Auto-correlation function Cross-correlation function How to model: Auto-regressive model Auto-regressive distributed lag model (lags of dependent and independent variables) = all models of this type are equilibrium-correction! = Estimate long-run mean in-sample and correct towards it! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

51 Forecasting Forecasting easy to implement but hard to do well (especially about the future...)! h-step forecasts (use real observations to update) Dynamic forecasts (use forecasted values to update) Do not evaluate a model by its forecast performance! Good models can forecast badly (or well...) Bad models can forecast well (or badly...) no connection between forecast performance and model validity Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

52 Non-Stationarity and Model Selection The world is non-stationary things change all the time! (think back 50 years ago) Many forms, some tameable: Trends: beware of spurious correlation! add linear trend to model! or take differences! Structural breaks: detect & correct for (dummy variables) Model Building and Model Selection Need to use your judgement and expertise! Can use automated methods (Autometrics) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

53 Putting it all together... Large set of tools - pick & mix but should follow coherent structure! Project Structure Example: Environmental Kuznets Curve (from Problem Sets) 1) Introduction Introduce the theory: Environmental Kuznets curve suggests inverse U-shape between income and pollution Why? - transition of developing economies Previous literature? Investigate using per capita CO2 emissions and per capita GDP for Japan Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

54 Data 2) Data per capita CO2 emissions (World Bank, in kg) per capita real GDP (FRED, in 2011 USD) theory suggests proportional relationship: log transform plot the data (raw and in logs) time series: in levels and differences scatter plot: CO2 against GDP Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

55 Data Plots 2) Data (continued) kg 5.0 CO2 per capita Real GDP per capita log(co2 kg) log(co2 kg) USD (2011) 2.5 log(co2) 10.5 log(rgdp) log(co2) log(usd) log(usd) log(rgdp) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

56 Methodology 3) Methodology Theory: inverse U-shape, requires quadratic functional form: log(co2) t = β 1 + β 2 log(rgdp) t + β 3 log(rgdp) 2 t + ɛ t Before we estimate, determine the time series properties: Partial (auto-correlation function) suggests 1-lag of log(co2) We also report a forecasting scenario and model in first differences to account for potential trends: Forecast from 1991 onwards First difference model: log(co2) t = β 1 + β 2 log(rgdp) t + β 3 log(rgdp) 2 t Use Automatic Model Selection to improve on theory model: include more lags and log(rgdp) t (economic growth) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

57 Results 4) Results Present model estimates in table incl. coefficients, standard errors, #observations, diagnostic tests for each model! plot fitted values and residuals! Table: Modelled Variable: log(co2 Emissions per capita) Variable Estimated Coefficient Constant (4.15)** L(RGDP) 9.32 (0.85)** L(RGDP) (0.044)** R T 51 AR 1-2 Test [0.00]** ARCH 1-1 Test [0.00]** Normality Test [0.50] Hetero Test [0.002]** RESET Test [0.00]** Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

58 4) Results continued: 2.25 log(co2 per capita) Fitted Model Residuals 1-step Forecasts 2.30 log(co2 per capita) log(co2 kg) SD 0 log(co2 kg) Label the axes! Label your data! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

59 4) Results continued: Automatic Model Selection Results general model (incl. lags and economic growth) reduced to: log(co2) = (0.0219) T=49 R 2 =0.987 log(co2) t (0.206) AR 1-2 test: F(2,44) = [0.1448] ARCH 1-1 test: F(1,47) = [0.3569] Normality test: Chiˆ2(2) = [0.2452] Hetero test: F(4,44) = [0.4854] Hetero-X test: F(5,43) = [0.6360] RESET23 test: F(2,44) = [0.1415] log(rgdp) t (0.0495) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

60 Discussion 5) Discussion Interpret model results: estimated 9.32% increase in emissions for 1% increase in GDP, diminishing marginal effect: -0.44% t-tests (individual) and F-tests (joint significance) ˆβ 2 > 0, ˆβ 3 < 0, consistent with EKC Estimates imply turning point of 34,500 USD ( 2007 in Japan) BUT: Model mis-specified, Kuznets theory not sufficient to explain CO2 emissions Automatic model selection: well specified model reduced to high persistence and economic growth effect suggests little evidence of Kuznets curve Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

61 Conclusion & References 6) Conclusion Summarize project: investigated EKC Mixed evidence of EKC for Japan, Japan past turning point But more likely that economic growth effect holds Further research: other countries, more variables 7) References Cite all data sources and referenced literature! e.g.: Stern, D. I. (2004). The rise and fall of the environmental Kuznets curve. World Development, 32 (8), avoid plagiarism! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

62 Principles of Modelling Start general and work to specific: Too few variables: Bias, cannot trust any statistics. Too many: too many to cope with? General: Can encompass many theories and test each one. Solution to problem of too many variables: Omit insignificant variables! If t-statistics are very small, omit variables. But time consuming and we can make mistakes. Can use automatic model selection. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

63 Potential Topics Environmental Kuznets Curve: look at other countries and extend the analysis. When are their estimated turning points? Purchasing Power Parity: investigate whether purchasing power parity holds for given exchange rates between countries Efficient market hypothesis: is it possible to model/predict share prices? The Phillips curve: theory suggests a link and trade-off between unemployment and inflation, test this using econometric methods. Econometric tools can be useful in many fields: temperatures and greenhouse gases temperatures and sea level policy effectiveness effect of interventions?... Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

64 Last thoughts Plot the data before you conduct any analysis! Have a clear structure in mind! (as in example) Reference all your sources!...remember the deadline: 5. June 2016 Building a sensible model with real data is very hard - don t be disheartened! me with any questions! felix.pretis@nuffield.ox.ac.uk Stay in touch and build models! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

65 Appendix Optional Appendix If you would like to learn more about how to deal with non-linear but rather stochastic trends and unit-roots, the next slides introduce the concept of cointegration! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

66 Re-Cap Yesterday: Stationary is the exception not the norm! Types of Non-stationarity: like going to zoo to look at non-elephants... Many many forms... = Three common forms encountered and tame-able: 1: Linear Trends 2: Structural Breaks (sudden change in parameters) 3: Unit-roots/Stochastic trends (advanced) can induce spurious correlation! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

67 Spurious Relationship: Example Japanese GDP and Cumulative Rainfall in Fortaleza (NE Brazil) Japan RGDP 6000 Cumulative Rainfall Fortaleza Brazil Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

68 Unit-Root Tests - Japanese RGDP: Unit-root tests The sample is: (38 observations and 2 variables) Jap_rGDP_90_billionY: ADF tests (T=34, Constant; 5%= %=-3.64) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob Cumulative Rainfall: cumrainfall_brazil: ADF tests (T=34, Constant; 5%= %=-3.64) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob = both series unit-root non-stationary Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

69 Model RGDP using Rainfall Japan RGDP Model Fit Residuals Model Jap.RDGP t = (4938) (1.45) cumrainfall Brazil t R 2 =0.98, F(1,36) = 2206 [0.000]** Cumulative Rainfall highly significant, explains 98% of variation in Japanese GDP Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

70 Addressing Spurious Relationships How can we actually estimate meaningful economic relationships? Method 1: Take differences of our data: If Y t is random walk, Y t = Y t Y t 1 is stationary. Regress Y t on X t. But we lose levels information. Economic theory in levels. Y t = (2976) (18.29) cumrainfall Brazil t + ˆɛ t. (9) = not significant anymore! Method 2: Cointegration Both series non-stationary Is a linear combination of them stationary? Residuals of regression of Y t on X t stationary: Test of cointegration. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

71 Cointegration: Drunken walk with dog Random Walk Ɛ 10 Random Walk Ɛ 1 Ɛ 5 Ɛ Stationary Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

72 Residuals Non-Stationary: 2 Residuals (scaled) Stationary: 2 Residuals Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

73 Cointegration Fundamental concept in econometrics: Engle and Granger won Nobel Prizes. Static regression model: Y t = β 1 + β 2 X t + ɛ t. (10) If Y t I(1) and X t I(1) (both non-stationary) expect combination non-stationary. ɛ t = Y t β 1 β 2 X t so expect ɛ t I(1), non-stationary. Hence residuals from Japan GDP and Brazilian rainfall are I(1). 2 Residuals (scaled) But if ɛ t I(0), stationary, then we have cointegration: Both series move together in a stationary economic relationship. 2.5 r:s (scaled) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

74 Cointegration exists if: Both Y t and X t are non-stationary and have unit roots. Combination of Y t and X t, Y t β 1 β 2 X t is stationary. Then we say: Y t and X t are cointegrated. If Y t, X t cointegrated then ɛ t = Y t β 1 β 2 X t stationary Engle & Granger test for cointegration: Unit root test on residuals. 1 Test Y t and X t for unit root. 2 2 If both have unit roots, run regression of Y t on constant and X t (static, don t include lags). 3 Save residuals ˆɛ t and run unit root test on them. 2 From Descriptive Statistics after clicking. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

75 Cointegration Examples Japanese RGDP (Y t ) and Cumulative Rainfall in Fortaleza Brazil (X t ) 1) Test Y t, X t for unit-roots to establish if I(1) - Japanese RGDP: Unit-root tests The sample is: (38 observations and 2 variables) Jap_rGDP_90_billionY: ADF tests (T=34, Constant; 5%= %=-3.64) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob Cumulative Rainfall: cumrainfall_brazil: ADF tests (T=34, Constant; 5%= %=-3.64) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob = both series unit-root non-stationary Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

76 2) Regress Y t on X t, store the residuals ˆɛ t Model Jap.RDGP t = (4938) (1.45) cumrainfall Brazil t Keep Residual: ˆɛ t = Y t ˆβ 1 ˆβ 2 X t ˆɛ t = Y t X t 2 Residuals (scaled) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

77 Test ˆɛ t for unit-root non-stationarity using Dickey-Fuller Test: ˆɛ t = ρˆɛ t 1 + v t Test H0: ρ 1 = 0 ˆɛ t = (ρ 1)ˆɛ t 1 + v t Unit-root tests The sample is: (38 observations and 1 variables) residuals: ADF tests (T=34, Constant; 5%= %=-3.64) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob Test Statistic = for no lags, for 1 lag Critical value = for 5% > 2.95 Cannot reject H0: Unit root = residual non-stationary, no cointegration Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

78 US Short-term and long-term interest rates US short-term (3-month inter-bank) (Y t ) and long-term (10-year bonds) (X t ) interest rates from 1964Q3 2013Q2 In short-run expect random walks Stable relationship in long-run? cointegration? 15 Short Term_US Long Term_US DShort Term_US DLong Term_US Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

79 Order of Integration of Levels? Determine Order of Integration: I(0)? I(1)?... Univariate Unit Root Tests on Y t, X t - US Short-Term Interest Rate The dataset is: new11.in7 The sample is: 1965(4) (2) (195 observations and 4 variables) ST_US: ADF tests (T=191, Constant; 5%= %=-3.47) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob US Long-Term Interest Rate LT_US: ADF tests (T=191, Constant; 5%= %=-3.47) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob = Unit-root non-stationary! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

80 Order of Integration of First Differences? Determine Order of Integration: I(0)? I(1)?... Univariate Unit Root Tests on First Differences Y t, X t - US Short-Term Interest Rate DST_US: ADF tests (T=191, Constant; 5%= %=-3.47) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob ** ** ** ** US Long-Term Interest Rate DLT_US: ADF tests (T=191, Constant; 5%= %=-3.47) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob ** ** ** ** Y t, X t Not Unit-root non-stationary! = Y t, X t are I(1): stationary if differenced once. Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

81 Static Regression 2) Regress Y t on X t, store the residuals ˆɛ t Model Short Term t = 1.75 (0.29) (0.04) Long Term t Keep Residual: ˆɛ t = Y t ˆβ 1 ˆβ 2 X t ˆɛ t = Y t X t 5.0 Short-Term Model Residuals Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

82 Test ˆɛ t for unit-root non-stationarity using Dickey-Fuller Test: ˆɛ t = ρˆɛ t 1 + v t Test H0: ρ 1 = 0 ˆɛ t = (ρ 1)ˆɛ t 1 + v t Unit-root tests The sample is: 1965(4) (2) (196 observations and 1 variables) US_ST_LT_residuals: ADF tests (T=191, Constant; 5%= %=-3.47) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC F-prob ** ** ** ** ** Test Statistics below critical values (-2.88 for 5%, for 1%) Reject H0: Unit root = residual stationary, series cointegrate! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

83 Interpretation: Short-run: Short-term and Long-term interest rates behave as random walks Long-run: Stationary relationship between them, never drift too far apart! Equilibrium Correction: e.g. Long-term interest rate changes, short-term interest rate adjusts to new equilibrium Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

84 Practical Computer Lab 5: Test for unit-root non-stationarity Test for cointegration Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

85 Short-Term and Long-Term Interest Rates Link between UK Short-term and Long-term interest rates Load data: interest rates.in7 Plot the data in levels and first differences (construct) Test for order of integration/unit-root non-stationarity 15 LT_UK ST_UK Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

86 Test for cointegration Estimate static regression model LT t = β 1 + β 2 ST t + ɛ t. Store the residual Test for unit-root non-stationarity on the residual 2.5 ST_UK_residuals Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

87 Cointegration between Stock Prices Closing price for Toyota, Honda, Ford load data car closing price.in7 Yesterday: found closing prices to be random walks Do the series cointegrate? = test for cointegration! Felix Pretis (Oxford) Time Series Akita Intl. University, / 88

88 Cointegration between Stock Prices Steps: Test for order of integration/unit root non-stationarity Estimate static regression model & store residual Test for cointegration (unit-root test on residual) Felix Pretis (Oxford) Time Series Akita Intl. University, / 88