2b Multivariate Time Series

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1 2b Multivariate Time Series Andrew Harvey Faculty of Economics, University of Cambridge May 2017 Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Multivariate time series and stationarity If the N 1 vector y t = (y 1t,.., y Nt ) is a weakly stationary time series with mean µ the autocovariance matrix at lag τ is given by Γ (τ) = E [ (y t µ) (y t τ µ) ], τ = 0, ±1, ±2,... The covariance matrix is Γ (0). This matrix is symmetric, but Γ (τ) is not generally symmetric at non-zero lags. However, it is always the case that Γ (τ) = Γ ( τ), τ = 1, 2, 3,... (1) The diagonal elements of γ (τ) are the autocovariances of the individual series. The off-diagonal elements are the cross-covariances. If y it and y jt denote the i-th and j-th elements in y t, the cross-covariance between the i-th and j-th series at lag τ is [ )] γ ij (τ) = E (y it µ i ) (y j,t τ µ j, τ = 0, ±1, ±2,... from which it is easily shown that γ ij (τ) = γ ji ( τ) as required by (1). Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

2 Multivariate ARMA processes The multivariate ARMA(p,q) model is y t = φ 1 y t φ p y t p +ξ t +θ 1 ξ t θ q ξ t q where ξ t is an N 1 vector of serially uncorrelated random variables with mean vector zero and covariance matrix Σ, that is multivariate white noise. The matrices φ and θ are N N matrices of autoregressive and moving average parameters. We can write φ (L) y t = θ (L) ξ t where and φ (L) = I φ 1 L φ p L p θ (L) = I + θ 1 L+ + θ q L q Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Vector autoregressive process -VAR(p) Atheoretical Sims (Econometrica, 1980) Number of parameters in a VAR (p) is N 2 p [+N (N + 1) /2 in Ω] since y t = φ 1 y t φ p y t p + ξ t but estimation easy as ML is just multivariate LS Example VAR(1) φ = [ ] TΣ 1 TΣ y t 1y t 1 y t 1y t t=2 t=2 Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

3 Multivariate ARIMA(p,d,q) and Seemingly Unrelated Time Series Equations (SUTSE) In the multivariate ARIMA(p, d, q) model d y t is multivariate ARIMA(p, q). Sometimes called VARIMA(p, d, q).. The simplest SUTSE model is y t = µ t +ε t, ε t WN(0,Σ ε ), t = 1,..., T, µ t = µ t 1 +η t, η t WN(0, Σ η ), where y t, µ t, ε t and η t are all N 1 vectors so that, for example, y t = (y 1t, y 2t,..., y Nt ). SSF is immediate. Initialization µ 1 = y 1 with MSE ( µ 1 ) = Σ ε. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Seemingly unrelated time series equations (SUTSE) Taking first differences gives The autocovariance matrices are: y t = η t + ε t ε t 1 Γ (0) = Σ η + 2Σ ε Γ (1) = Σ ε = Γ ( 1) Γ (τ) = 0, τ 2 The condition for strict invertibility of the stationary form is that Σ η should be p.d. Compare with ACF of VMA(1) - y t = ξ t + Θξ t 1. Fewer parameters in SUTSE and invertibility easily imposed. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

4 Data Irregularities: forecasting and nowcasting with auxiliary series The use of an auxiliary series that is a coincident or leading indicator yields potential gains for nowcasting and forecasting. Take one series to be the target series while the second is the related series. With nowcasting our concern is with the reduction in the MSE in estimating the level and the slope. This translates into gains for forecasting. The SSF allows series with different frequencies to be modelled together. Frale et al (2010, JRSS) Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Forecasting and nowcasting with auxiliary series Labour Force Survey (Harvey and Chung, 2000, JRSS A) - Combine quarterly survey data on unemployment with the monthly claimant count. The model for the monthly CC series is a local linear trend with no irregular component. The monthly model for the LFS series is similar, except that the observations contain a survey sampling error earlier. A bivariate model with these features can be handled within the state space framework even if the LFS observations are only available every quarter or, as was the case before 1992, every year. The underlying trends in the two series are not the same. However, such divergence does not mean that the CC series contains no usable information. For example it is plausible that the underlying slopes of the two series move closely together even though the levels show a tendency to drift apart. There is a considerable gain in the precision with which the underlying change in ILO unemployment is estimated. Models were estimated using monthly CC observations from 1971 together with quarterly LFS observations from May 1992 and annual observations from The weighting functions are shown below Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

5 (Thousands) Figure 1. LFS Unemployed and Claimant Count for Great Britain (Seasonally Ad Claimant Count LFS ILO Unemployed Figure: Annual and quarterly observations from the British labour force survey and the monthly claimant count Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Bivariate model for UC Phillips curve Rather than first estimating the output gap from a univariate model for GDP, inflation and GDP may be modelled jointly as [ ] [ ] [ ] [ ] πt µ π = t ψ π + t ε π y t ψ y + t t ε y (2) t µ y t where µ π t is a random walk and µ y t is an integrated random walk. These two stochastic trends are independent of each other. The STAMP 8 package allows the random walk and integrated random walk restrictions to be imposed on the model. Similar cycles. The irregular disturbances may be correlated with each other. A seasonal component can be added. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

6 0.075 INFLATION Lev el log_gdp Lev el INFLATION Lev el+cycle log_gdp Cycle Figure: Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Common trends and co-integration A common factor model contains stochastic components with fewer than N elements. Recognition of common factors yields models which may not only have an interesting interpretation, but may also provide more effi cient inferences and forecasts. In terms of a SUTSE model, the presence of common factors means that the covariance matrices of the relevant disturbances are less than full rank. Common factors in the trend imply co-integration. At its simplest co-integration means that there exists a linear combination of two I (1) series that is stationary, ie I (0). If the series are I (2), there is co-integration if there is a linear combination that is I (0) or I (1) - different orders of co-integration. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

7 Common trends and co-integration The basic ideas can be illustrated with a bivariate local level model y 1t = µ 1t + ε 1t, µ 1t = µ 1t 1 + η 1t, t = 1,..., T, y 2t = µ 2t + ε 2t, µ 2t = µ 2t 1 + η 2t The covariance matrix of (η 1t, η 2t ) may be written [ ] σ 2 Σ η = 1η ρ η σ 1η σ 2η ρ η σ 1η σ 2η where ρ η is the correlation. σ 2 2η Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Common trends and co-integration The model can be transformed as follows: y 1t = µ 1t + ε 1t, (3) y 2t = πµ 1t + µ t + ε 2t, where π = ρ η σ 2η /σ 1η and the covariance matrix of the disturbances driving the new multivariate random walk (µ 1t, µ t ) is Var [ η1t η t ] = [ σ 2 1η 0 0 σ 2 2η ρ2 η σ2 2η ]. In other words by setting η 2t = πη 1t + η t, two uncorrelated levels, µ 1t and µ t, based respectively on η 1t and η t, are obtained. (In a Gaussian model, these expressions can be thought of as coming from the expression for the mean and variance of η 2t, conditional on η 1t ). Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

8 Common trends and co-integration If ρ η = ±1, then µ t is constant and there is only one common trend which will now be denoted as µ t rather than µ 1t. Hence with y 1t = µ t + ε 1t, t = 1,..., T, y 2t = πµ t + µ + ε 2t µ t = µ t 1 + η t If π = 1, the trend in the second series is always at a constant distance, µ, from the trend in the first series, that is µ 2t = µ + µ 1t. This is known as balanced growth. ( When the series are modelled in logarithms, the level of the first trend must be multiplied by exp(µ) to give the second.) If µ = 0 the two trends are identical. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / log_cons log_gdp log_inv Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

9 Common trends and co-integration Pre-multiplying the observation vector in (3) by the vector ( π, 1) gives y 2t = πy 1t + µ t + ε t (4) where ε t = ε 2t πε 1t. When ρ η = ±1, the linear combination y 2t πy 1t is stationary and so the series are co-integrated. The full model can be written as y 1t = µ t + ε 1t, (5) y 2t = πy 1t + µ + ε t The first equation generates a nonstationary series for y 1t, while the second equation is the co-integrating relationship. This form of the model may be estimated directly. Equation (4) suggests that a test of the null hypothesis that the two series are co-integrated amounts to testing whether µ t is constant against the alternative that it is a random walk. If π is given this is a standard stationarity test. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 BECM Setting the model out as in (5) emphasises the co-integrating relationship in the model. More generally any bivariate system with a co-integrating may be represented by the triangular form Co-integration may also be characterised in terms of the moving average representation of the differenced observations. If a finite autoregressive representation is possible, it is most conveniently expressed as a VECM. The simplest such model for two series (BECM) is y 1t = γ 1 (y 1,t 1 αy 2,t 1 µ ) + ϕ 11 y 1t 1 + ϕ 12 y 2t 1 + ξ 1t y 2t = γ 2 (y 1,t 1 αy 2,t 1 µ ) + ϕ 21 y 1t 1 + ϕ 22 y 2t 1 + ξ 2t where ξ 1t and ξ 2t are multivariate white noise. Duality with common trends. Leads to different approaches to testing. Serial correlation is handled by adding lagged differences of all the variables to each equation - similar to ADF. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

10 Balanced growth The balanced growth UC model is a special case of (??): y t = iµ t +µ c +ψ t +ε t,, t = 1,..., T, where µ t is a univariate local linear trend, i is a vector of ones, and µ c is an N 1 vector of constants, subject to constraints so it contains only N 1 free parameters, eg one may be set to zero. Although the levels may be different, the slopes are the same, irrespective of whether they are fixed or stochastic. A balanced growth model implies that the series have a stable relationship over time. This means that there is a full rank (N 1) N matrix, D, with no null columns and the property that Di = 0, thereby rendering Dy t stationary. The rows of D may be termed balanced growth co-integrating vectors. Typically each row will contain a one, a minus one and zeroes elsewhere. For example, one country may be used as a benchmark. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Intervention analysis and control groups The bivariate local level model is y 1t = µ 1t + ε 1t, t = 1,..., T, y 2t = µ 2t + λw t + ε 2t, where the first series contains the observations on the control group. It is assumed that the intervention occurs at some point during the sample, ie 1 < τ < T. The higher the correlation between the groups, the greater the gain in precision with which the intervention effect may be estimated. A control group which is co-integrated with the experimental group is likely to be very valuable since it enables a consistent estimator to be constructed. (6) Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

11 Intervention analysis and control groups When the two series are driven by a common level, and therefore co-integrated, the second equation becomes y 2t = θµ t + µ + λw t + ε 2t, t = 1,..., T, The two equations can be estimated jointly by ML. Alternatively the model may be transformed to with y 1t = µ t + ε 1t, y 2t = θy 1t + µ + λw t + ε t, t = 1,..., T. (7) The second equation is the co-integrating relationship with the intervention included. Estimating these two equations together is equivalent to estimating the original two equations by ML. However, it has the technical advantage that, because y 1t is an explanatory variable in (7), θ may be concentrated out of the likelihood function. Furthermore its standard error is readily available. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Seat belts in GB and inflation targeting in Sweden The intervention analysis of the effect of the British seat belt law can be extended by using control groups. To be more specific, rear seat passengers will be used as a control for front seat passengers since the 1983 law did not require those in the rear to wear belts. See Harvey (International Statistical Review,1996). STAMP manual. Are rear seat passengers a valid control group? Risk compensation. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

12 Drivers Level+Intv Rear Level Figure: Smoothed levels of Drivers and Rear Seat Passengers killed and seriously injured in Great Britain, with allowance made for the seat belt law. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Seat belts in GB (1) Estimate of λ in a univariate model is with t = 3.59 (2) Bivariate - correlation between level disturbances is Estimate of λ is with t = 6.17 (3) Balanced growth - estimate of λ is with t = 9.25 Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

13 Control groups: Examples Example The speed limit on some U.S. rural highways was relaxed in A model fitted to monthly data on crashes in rural Arizona, with the intervention in April The t-statistic on the intervention was 2.21, but a joint model with the series on crashes on urban highways, where the speed limit was unchanged, increased the t-statistic to The gain arose from the correlation of 0.81 in the level disturbances. See Balkin, S. and J.K. Ord (2001). Assessing the impact of speed-limit increases on fatal interstate crashes (with discussion). Journal of Transportation and Statistics, 4, Example Sweden adopted inflation targetting but EMU did not. See Angeriz, A. and Arestis, P. (2008). Oxford Economic Papers, 60, Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28 Dynamic intervention effects Assuming balanced growth, y 0t = µ t + µ 0 + λd t + m j=1 λ j d t + ε 0t, t = 1,..., T, y t = iµ t + µ + ε t, where the permanent shift in the level of y 0t is captured by a step dummy d t = { 0 for t < τ + m, 1 for t τ + m, 1 < τ + m T, whereas the intermediate effects are modeled by m pulse dummies d t = { 0 for t = τ + j 1, 1 for t = τ + j 1, j = 1,..., m. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28

14 Dynamic effects Growing literature on synthetic control and panel control. Abadie, A., Diamond, A., and J. Hainmueller. (2010) Synthetic control methods for comparative case studies: Estimating the effect of California s tobacco control program. Journal of the American Statistical Association, 105, Abadie, A., Diamond, A., and J. Hainmueller. (2014) American Journal of Political Science, 59, Gardeazabal, J. and A. Vega-Bayo (2016) An Empirical Comparison between the Synthetic Control Method and Hsiao s et al s Panel Data Approach to Program Evaluation. Journal of Applied Econometrics (forthcoming). Harvey, A.C. and S. Thiele. (2017). Co-integration and Control. Mimeo. Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / California Colorado CalSmo Montana Andrew Harvey (Faculty of Economics, University of Cambridge) 2b Multivariate Time Series May / 28