时间序列分析及VAR模型Word下载.docx

1、一，+ _i + f2tTlie beauty of tliis model is that we dont need to predefine whether x or y are endogenous （the dependent variable）. In fact, we can test whether x （y） is endogenous or exogenous using Granger causality tests The idea of Granger causality is that past observations （lagged dependent varia

2、bles） can influence cuiTent observations 一 but not vice versa So the idea is rather simple: the past affects the present, and the present does not affect the past. ST AT A provides Granger causality tests after conducting a VAR analysis, which is based on testing the joint hypothesis that past reali

3、sations do not Granger cause the present realisation of the dependent variable.In many applications, VAR models make a lot of sense, as a clear direction of causality cannot be predefined For instance, there is a substantial literature on the benefits of internationalisation （e.g. entering foreign m

4、arket through cross-border M&A）. There is evidence that multinationals outperform local peers due to the benefits of operating in many countries At the same time, we know that high-performing companies are more likely to enter foreign markets due to their ownership specific advantages This argument

5、is based on the Resource-based View and the OLS framework developed by Dunning and Rugman （Reading School of International Business）. The VAR model allows you to incorporate both effects: in fact you can test whether performance drives internationalisation or internationalisation drives performance.

6、Before you start using a VAR modeL you have to make sure that the time series are stationary. So the first step is to check whether the time series is sStionjry using Dickey-Fuller tests and KPSS tests. The second step is to specify the optimal lag length （p） of the model. This is done by comparing

7、different model specifications using information criteria. Apart from using Akaike （AIC） and Bayesian Schwarz （BIC）, the Hannan-Quinn （HQIC） is commonly used. Most applied econometricians favour the Hannan-Quinn （HQIC） criterion STATA will help you to make a good choice* After specifying your model,

8、 you need to check stability conditions. The coefficient matrix of the reduced form VAR has to ensure that the iteration sequence converges to a long-term value STATA will help you in checking stability.To be precise, you need to show that the eigenvalues of the coefficient matrix lie within the uni

9、t circle The reason behind it can be only understood when you understand the method of diagonalizing a matrix.VAR models offer ano 什 nice feature: impulse response functions VAR models capture the dynamics of two （or more） stationary time series; hence, we can assess the dynamic impact of a marginal

10、 change of one variable on another The standard OLS regression provides coefficients, and coefficients refer to the partial impact of an explanatory variable on the dependent variable In the case of VAR models, the relationship becomes dynamic, as a change of one variable（say x） in t can affect x an

11、d y in t+1 The impact on x and y in t+l in turn affects x and y in t+2 and so on until the impact dies out. Impulse response functions are very useful in illustrating the short-term dynamics in a model.Lefs look at an example to see how VAR modelling works In Lecture 5, we tried very hard to underst

12、and gold prices We extend our univariate model by exploring the relationships between gold and silver prices Linking two （similar） assets or securities is a veiy common trading strategy, which is called pairs-tradin Before we do any sophisticated modelling, it is always beneficial to look at some li

13、ne charts. Figure 1 shows the indexed time series of nominal gold and silver prices from 1900 to 2010.Figure 1: Nominal gold and silver prices, indexedA 1900-2010We can see that there is a certain degree of co-movement, which we might be able to exploit for our trading strategy. Before we can use VA

14、R. we need to ensure that both time series are stationary. It is obvious from Figure 1 that gold and silver prices are not stationary. However, after taking a first-difference we can show that price changes are stationary So both time series are 1（1）.The next step is to determine the optimal lag len

15、gth using information criteria Table 1 shows different specifications using the varsoc commandTable 1: Determining the optimal lag length using information criterialagLLLRdfPFPEAICHQICSBIC104 118 00049 1.94511丄92463JL- 894561113.551JI R- 86640 00 l 0004422- 0486-1.98714JI 89694*2120.01512.9270.01? 0

16、0042? ifK-2.09552*1.9931*-1.R42763120- ft191- 60860- 807 00044R2.03465 : I 89126-1.6R079124- R268.01360 09 l 00044ft-2.03478-1.85042-1.5798?5130.12610.599*0.031 000438-2.05954-1.83421-1.50347Number of obsEndogenous: return.g returns Exogenous: _consSelectioreorder criteriaSample: 1906 - 2010105Based

17、 on the AIC and HQIC, two lags are optimal; however, the （S）BIC prefers only one lag. I would prefer HQIC and try two lags first. If the second lag does not exhibit significant coefficient, we could try to reduce the lag leng 什】in line with （S）BIC-We run a VAR with two lags to explain current price

18、changes in gold and silver- Table 2 provides the OLS estimates.Table 2: VAR model with two lags 1903 - 2010I I SI # I : 1 X% 1d no AdOO 0004Det（Sigma_ml）二 OOOM323NO- of obs AIC=108 =2 148455 二 2 04776 = 丄 90OilParmsFcmm lienreturn.g 5RMSE 126927 0.2425chi2 Pchi234.5786 0.0000Vector autoregressionret

19、urns5 196569 0.1306 16.22763 0- 0027coef.Std Err.P|Z|95% Conf.Intervalreturngreturn.a11- 4ftR4in7 l 29QREA2QAn non 7274SR1L2 -0139809 122817-0.110- 909-.2546979 2267361return.sI : I-006R1266R05903-0.0R0.933-164766R 1511415L2 207786 0807151-2.570.010-.3659847 0495874_cons 0277213 01248572.22O 026 003

20、2497 0521929returns returnAg如427弘 1Q04R4ft1 ASn OQQ-0SRQ9S7 AR7AR9A 1085011 19020380.570.568-2642915 4812937 1094293124B0阳0RR0.3R1-.1351905 354049-3201805 1250015-2.56-.5651789 0751821 024511.01933631.270- 205-.0133875 0624095We see that silver prices （lag 2） affect current gold prices, and we can e

21、stablish autocorrelation in both time series* To test whether gold Granger causes silver or vice versa, we run Granger causality tests reported in Table 3.Table 3: Granger causality testsGranger causality Wald testsEquationExcludedchi 2df Prob chi2return_greturns6.7650.034ALL3- 96150.138Hence, we co

22、nfirm that past changes in silver prices can predict future gold price changes. This is very interesting, as it can be used to develop a trading strategy. Finally, we need to show that the VAR is stable （see Table 4）.Table 4: Stability condition of the VAREigenvalue stability conditionEigenvalueModu

23、lus 2367286 + 362415?.432882367286 - 3624157 43288 06119136 + 3747777?37974 06119136- 37477771 37974All the eigenvalues lie in side the unit circle VAR satisfies stability conditio nFinally, we can illustrate the impact of silver price changes on future gold price changes using an impulse response f

24、unction* Figure 2 shows the impulse response function and confidence intervals derived from bootstrapping. If silver prices increase today by 1%, we should expect a significant decline in gold prices in two years by 0.2% Figure 2: Impulse response functionr, return_s, return_g 2 0 2 4step95% Cl impulse response f