# 3.1 Stationary Processes and Mean Reversion

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 3. Univariate Time Series Models 3.1 Stationary Processes and Mean Reversion Definition 3.1: A time series y t, t = 1,..., T is called (covariance) stationary if (1) E[y t ] = µ, for all t Cov[y t, y t k ] = γ k, for all t Var[y t ] = γ 0 (< ), for all t Any series that are not stationary are said to be nonstationary. Stationary time series are mean-reverting, because the finite variance guarantees that the process can never drift too far from its mean. The practical relevance for a trader is that assets with stationary price series may be profitably traded by short selling when its price is above the mean and buying back when the price is below the mean. This is known as range trading. Unfortunately, by far the most price series must be considered nonstationary. 1

2 When deriving properties of time series processes, we shall frequently exploit the following calculation rules, where small and capital letters denote constants and random variables, respectively. Expected Value: (2) (3) (4) Variance: (5) E(aX + b) = ae(x) + b E(X + Y ) = E(X) + E(Y ) E(XY ) = E(X)E(Y ) for X, Y indep. V (X) = E(X 2 ) E(X) 2 (6) V (ax + b) = a 2 V (X) (7) V (X +Y ) = V (X)+V (Y )+2Cov(X, Y ) Covariance: (8) Cov(X, Y ) = E(XY ) E(X)E(Y ) Cov(X, Y ) = Cov(Y, X), Cov(X, X) = V (X) and for independent X, Y : Cov(X, Y ) = 0 (9) Cov(aX + b, cy + d) = ac Cov(X, Y ) (10) Cov(X +Y, Z) = Cov(X, Z)+Cov(Y, Z) 2

3 Stationarity implies that the covariance and correlation of observations taken k periods apart are only a function of the lag k, but not of the time point t. Autocovariance Function (11) γ k = Cov[y t, y t k ] = E[(y t µ)(y t k µ)] k = 0, 1, 2,.... Variance: γ 0 = Var[y t ]. Autocorrelation function (12) ρ k = γ k γ 0. Autocovariances and autocorrelations of covariance stationary processes are symmetric. That is, γ k = γ k and ρ k = ρ k (exercise). 3

4 The Wold Theorem, to which we return later, assures that all covariance stationary processes can be built using using white noise as building blocks. This is defined as follows. Definition 3.2: The time series u t is a white noise process if (13) E[u t ] = µ, for all t Cov[u t, u s ] = 0, for all t s Var[u t ] = σ 2 u <, for all t. We denote u t WN(µ, σ 2 u ). Remark 3.2: Usually it is assumed in (13) that µ = 0. Remark 3.3: A WN-process is obviously stationary. 4

5 As an example consider the so called first order autoregressive model, defined below: AR(1)-process: (14) y t = φ 0 + φ 1 y t 1 + u t with u t WN(0, σ 2 ) and φ 1 < 1. Note that the AR(1) process reduces to white noise in the special case that φ 1 = 0. We shall later demonstrate how the AR(1) process may be written as a linear combination of white noise for arbitrary φ 1 < 1 and show that this condition is necessary and sufficient for the process to be stationary. For now we assume that the process is stationary, such that among others E(y t ) = E(y t 1 ) and V (y t ) = V (y t 1 ) and use this to derive the expectation and variance of the process together with its first order autocorrelation coefficient. 5

6 Taking unconditional expectation and variance of (14) yields (15) E(y t ) = φ 0, V (y t ) = σ2 1 φ 1 1 φ 2. 1 In deriving the first order autocorrelation coefficient ρ 1, note first that we may assume without loss of generality that φ 0 = 0, since shifting the series by a constant has no impact upon variances and covariances. Then E(y t ) = E(y t 1 ) = 0, such that by independence of u t and y t 1 : γ 1 = Cov(y t, y t 1 ) = E[(φ 1 y t 1 + u t ) y t 1 ] = φ 1 E(y 2 t 1 ) + E(u t) E(y t 1 ) = φ 1 V (y t 1 ) = φ 1 γ 0. Hence (16) ρ 1 = γ 1 /γ 0 = φ 1. That is, the first order autocorrelation ρ 1 is given by the lag coefficient φ 1. 6

7 As the following slide demonstrates, mean reversion is the faster the lower the autocovariances are. The fastest reverting process of all is the white noise process, which may be interpreted as an AR(1) process with φ 1 = 0. The larger the the autocovariance becomes, the longer it takes to revert to the mean and the further drifts the series away from its mean despite identical innovations u t. In the limit when φ 1 = 1, the series is no longer stationary and there is no mean reversion. 7

8 Realisations from AR(1) Model rho = Realisations from AR(1) Model rho = Realisations from AR(1) Model rho =

9 The AR(1) process may be generalized to a pth order autoregressive model by adding further lags as follows. AR(p)-process: (17) y t = φ 0 + φ 1 y t 1 + φ 2 y t φ p y t p + u t where u t is again white noise with µ = 0. The AR(p)-process is stationary if the roots of the characteristic equation (18) m p φ 1 m p 1... φ p = 0 are inside the unit circle (have modulus less than one). Example: The characteristic equation for the AR(1)-process is m φ 1 = 0 with root m = φ 1, such that the unit root condition m < 1 is identical to the earlier stated stationarity condition φ 1 < 1 for AR(1)-processes. 9

10 The moving average model of order q is MA(q)-process: (19) y t = c+θ 1 u t 1 +θ 2 u t θ q u t q +u t where again u t W N(0, σ 2 ). Its first and second moments are (20) (21) (22) E(y t ) = c V (y t ) = (1 + θ1 2 + θ θ2 q )σ 2 γ k = (θ k + θ k+1 θ θ q θ q k )σ 2 for k q and 0 otherwise. This implies that in contrast to AR(p) models, MA(q) models are always covariance stationary without any restrictions on their parameters. 10

11 Autoregressive Moving Average (ARMA) model A covariance stationary process is an ARMA(p,q) process of autoregressive order p and moving average order q if it can be written as (23) y t = φ 0 + φ 1 y t φ p y t p +u t + θ 1 u t θ q u t q For this process to be stationary the number of moving average coefficients q must be finite and the roots of the same characteristic equation as for the AR(p) process, m p φ 1 m p 1... φ p = 0 must all lie inside the unit circle. EViews displays the roots of the characteristic equation in the ARMA Diagnostics View, which is available from the output of any regression including AR or MA terms by selecting ARMA Structure under Views. 11

12 Example: (Alexander, PFE, Example II.5.1) Is the ARMA(2,1) process below stationary? (24) y t = y t y t 2 + u t + 0.5u t 1 The characteristic equation is with roots m m = 0 m = 0.75 ± = 0.75 ± i = ± i, where i = 1 (the imaginary number). The modulus of both these roots is = 0.5, which is less than one in absolute value. The process is therefore stationary. 12

13 Inversion and the Lag Operator ARMA models may be succinctly expressed using the lag operator L: Ly t = y t 1, L 2 y t = y t 2,..., L p y t = y t p In terms of lag-polynomials (25) φ(l) = 1 φ 1 L φ 2 L 2 φ p L p (26) θ(l) = 1 + θ 1 L + θ 2 L θ q L q the ARMA(p,q) in (23) can be written shortly as (27) φ(l)y t = φ 0 + θ(l)u t or (28) y t = µ + θ(l) φ(l) u t, where (29) µ = E[y t ] = φ 0 1 φ 1 φ p. 13

14 It turns out that the stationarity condition for an ARMA process, that all roots of the characteristic equation be inside the unit circle, may be equivalently rephrased as the requirement that all roots of the polynomial (30) φ(l) = 0 are outside the unit circle (have modulus larger than one). Example: (continued). The roots of the lag polynomial Φ(L) = L L 2 = 0 are L = 3 ± = 1.5 ± 2 with modulus = 2 > i Hence the process is stationary. 14

15 Wold Decomposition Theorem 3.1: (Wold) Any covariance stationary process y t, t =..., 2, 1, 0, 1, 2,... can be written as an infinite order MA-process (31) y t = µ + u t + a 1 u t = µ + h=0 a h u t h, where a 0 = 1, u t WN(0, σ 2 u), and h=0 a 2 h <. In terms of the lag polynomial a(l) = a 0 + a 1 L + a 2 L 2 + equation (31) can be written in short as (32) y t = µ + a(l)u t. 15

16 For example, a stationary AR(p) process (33) φ(l)y t = φ 0 + u t can be equivalently represented as an infinite order MA process of the form (34) y t = φ(l) 1 (φ 0 + u t ), where φ(l) 1 is an inifinite series in L. Example: Inversion of an AR(1)-process The lag polynomial of the AR(1)-process is φ(l) = 1 φ 1 L with inverse (35) φ(l) 1 = Hence, (36) y t = i=0 i=0 = φ 0 1 φ 1 + φ i 1 Li for φ 1 < 1. φ i 1 Li (φ 0 + u t ) i=0 φ i 1 u t i. As a byproduct we have now proven that the AR(1) process is indeed stationary for φ 1 < 1, because then it may be represented as an MA process, which is always stationary. 16

17 Similiarly, a moving average process (37) y t = c + θ(l)u t may be written as an AR process of infinite order (38) θ(l) 1 y t = θ(l) 1 c + u t, provided that the roots of θ(l) = 0 lie outside the unit circle, which is equivalent to the roots of the characteristiq equation m q + θ 1 m q θ q = 0 being inside the unit circle. In that case the MA-process is called invertible. The same condition applies for invertibility of arbitrary ARMA(p,q) processes. Example: (II.5.1 continued) The process (24) y t = y t y t 2 + u t + 0.5u t 1 is invertible because θ(l) = L = 0 when L = 2 < 1, and m = 0 when m = 0.5, such that m < 1. 17

18 Second Moments of General ARMA Processes We shall in the following derive a strategy to find all autocorrelations for arbitrary ARMA processes. Consider the general ARMA process (23), where we assume without loss of generality that φ 0 = 0. Multiplying with y t k and taking expectations yields (39) E(y t k, y t ) = φ 1 E(y t k, y t 1 ) + + φ p E(y t k, y t p ) +E(y t k, u t )+θ 1 E(y t k, u t 1 )+ +θ q E(y t k, u t q ). This implies for k > q: (40) γ k = φ 1 γ k φ p γ k p. Dividing by the variance yields the so called Yule Walker equations, from which we may recursively determine the autocorrelations of any stationary ARMA process for k > q: (41) ρ k = φ 1 ρ k φ p ρ k p. 18

19 Example: Autocorrelation function for AR(1) We know from equation (16) that the first order autocorrelation of the AR(1) process is ρ 1 = φ 1. y t = φ 0 + φ 1 y t 1 + u t Hence, by recursively applying the Yule Walker equations with p = 1 we obtain (42) ρ k = φ k 1, such that the autocorrelation function levels off for increasing k, since φ 1 < 1. For the general ARMA(p,q) process (23), we obtain for k = 0, 1,..., q from (39) a set of q + 1 linear equations, which we may solve to obtain γ 0, γ 1,..., γ q, and the autocorrelations ρ 1,..., ρ q from dividing those by the variance γ 0. 19

20 Example: ACF for ARMA(1,1) Consider the ARMA(1,1) model (43) y t = φy t 1 + u t + θu t 1. The equations (39) read for k =0 and k =1: (44) (45) γ 0 = φγ 1 + σ 2 + θ(θ + φ)σ 2, γ 1 = φγ 0 + θσ 2. In matrix form: (46) ( 1 φ φ 1 such that (47) ( ) γ0 γ 1 ) ( ) γ0 = 1 1 φ 2 γ 1 = ( (1 + θ(θ + φ))σ 2 θσ 2 ) ( ) ( 1 φ (1 + θ(θ + φ))σ 2 φ 1 θσ 2 where we have used that ( ) 1 1 φ (48) = 1 φ 1 1 φ 2 ( ) 1 φ. φ 1, ), 20

21 The variance of the ARMA(1,1) process is therefore (49) γ 0 = 1 + 2θφ + θ2 1 φ 2 σ 2. Its first order covariance is (50) γ 1 = φ + θ2 φ + θφ 2 + θ 1 φ 2 σ 2, and its first order autocorrelation is (51) ρ 1 = (1 + φθ)(φ + θ) 1 + 2θφ + θ 2. Higher order autocorrelations may be obtained by applying the Yule Walker equations ρ k = φρ k 1. Note that these results hold also for ARMA(1,1) processes with φ 0 0, since variance and covariances are unaffected by additive constants. 21

22 Response to Shocks When writing an ARMA(p,q)-process in MA( ) form, that is, (52) y t = µ + ψ(l)u t, then the coefficients ψ k of the lag polynomial (53) ψ(l) = 1 + ψ 1 L + ψ 2 L measure the impact of a unit shock at time t on the process at time t + k. The coefficient ψ k as a function of k = 1, 2,... is therefore called the impulse response function of the process. 22

23 In order to find its values, recall from (28) that ψ(l) = θ(l)/φ(l), such that (54) 1 + θ 1 L θ q L q =(1 φ 1 L... φ p L p ) (1 + ψ 1 L + ψ 2 L ) Factoring out the right hand side yields (55) 1 + θ 1 L θ q L q =1 + (ψ 1 φ 1 )L + (ψ 2 φ 1 ψ 1 φ 2 )L 2 + (ψ 3 φ 1 ψ 2 φ 2 ψ 1 φ 3 )L such that by equating coefficients: (56) ψ 1 =θ 1 + φ 1 ψ 2 =θ 2 + φ 1 ψ 1 + φ 2 ψ 3 =θ 3 + φ 1 ψ 2 + φ 2 ψ 1 + φ

24 Example: (Alexander, PFE, Example II.5.2) Consider again the stationary and invertible ARMA(2,1) process (24) y t = y t y t 2 + u t + 0.5u t 1. The impulse response function is given by ψ 1 = = 1.25, ψ 2 = = ψ 3 = = ψ 4 = = EViews displays the impulse response function under View/ARMA Structure... /Impulse Response. 24

25 Forecasting with ARMA models With estimated parameters ˆµ, ˆφ 1,..., ˆφ p and ˆθ 1,..., ˆθ q, the optimal s-step-ahead forecast ŷ T +s for an ARMA(p,q)-process at time T is (57) ŷ T +s = ˆµ + p i=1 ˆφ i (ŷ T +s i ˆµ) + q j=1 ˆθ j u T +s j, where u T +s j is set to zero for all future innovations, that is, whenever s > j. These are easiest generated iteratively: ŷ T +1 = ˆµ+ ŷ T +2 = ˆµ+ and so on. p i=1 p i=1 ˆφ i (y T +1 i ˆµ)+ ˆφ i (ŷ T +2 i ˆµ)+ q j=1 q j=2 ˆθ j u T +1 j, ˆθ j u T +2 j, These are called conditional forecasts, because they depend upon the recent realizations of y t and u t, and are available from EViews (together with confidence intervals) from the Forecast button. 25

26 Unconditional Forecasting The unconditional (long-run) forecast for an ARMA(p,q)-process is given by the estimated mean (58) ˆµ = ˆφ 0 1 ˆφ 1 ˆφ p. y t is normally distributed with mean µ and variance γ 0, such that the usual confidence intervals for normally distributed random variables apply. In particular, a 95% confidence interval for y t is (59) CI 95% = [ ˆµ ± 1.96 ] ˆγ 0. This is of practical interest when the series y t represents an asset which can be traded and one wishes to identify moments of overor underpricing. 26

27 Example. Estimating an ARMA(1,1) model on the spread in Figure II.5.3 of Alexanders book (PFE) yields the output below: Dependent Variable: SPREAD Method: Least Squares Date: 01/18/13 Time: 13:35 Sample (adjusted): 9/21/2005 6/08/2007 Included observations: 399 after adjustments Convergence achieved after 9 iterations MA Backcast: 9/20/2005 Variable Coefficient Std. Error t-statistic Prob. C AR(1) MA(1) 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 Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots.87 Inverted MA Roots.60 ˆµ = and applying the variance formula for ARMA(1,1) processes (49) yields ˆγ 0 = ( 0.6)+( 0.6) = CI 95% = ± = [ 44.98; ] 27

28 Partial Autocorrelation The partial autocorrelation function φ kk of a time series y t at lag k measures the correlation of y t and y t k adjusted for the effects of y t 1,..., y t k+1. It is given by the k th order autocorrelation function of the residuals from regressing y t upon the first k 1 lags: (60) φ kk = Corr[y t ŷ t, y t k ŷ t k ], where (assuming here for simplicity φ 0 = 0) (61) ŷ t = ˆφ k1 y t 1 + ˆφ k2 y t ˆφ k 1,k 1 y t k+1. It may also be interpreted as the coefficient φ kk in the regression (62) y t = φ k1 y t 1 + φ k2 y t φ kk y t k + u t. From this it follows immediately that in an AR(p)-process φ kk = 0 for k > p, which is a help in discriminating AR-processes of different orders with otherwise similiar autocorrelation functions. 28

29 Estimation of acf Autocorrelation (and partial autocorrelation) functions are estimated by their empirical counterparts (63) ˆγ k = 1 T k where T t=k+1 ȳ = 1 T is the sample mean. (y t ȳ)(y t k ȳ), T y t t=1 Similarly (64) r k = ˆρ k = ˆγ k ˆγ 0. 29

30 Statistical inference If the model is well specified, the residuals must be white noise, thas is in particular uncorrelated. It can be shown that if ρ k = 0, then E[r k ] = 0 and asymptotically (65) Var[r k ] 1 T. Similarly, if φ kk = 0 then E[ˆφ kk ] = 0 and asymptotically (66) Var[ˆρ kk ] 1 T. In both cases the asymptotic distribution is normal. Thus, testing (67) H 0 : ρ k = 0, can be tested with the test statistic (68) z = T r k, which is asymptotically N(0, 1) distributed under the null hypothesis (67). 30

31 The Portmanteau statistics to test the hypothesis (69) H 0 : ρ 1 = ρ 2 = = ρ m = 0 is due to Box and Pierce (1970) (70) Q (m) = T m k=1 r 2 k, m = 1, 2,..., which is (asymptotically) χ 2 m - distributed under the null-hypothesis that all the first autocorrelations up to order m are zero. Mostly people use Ljung and Box (1978) modification that should follow more closely the χ 2 m distribution (71) Q(m) = T (T + 2) m k=1 1 T k r2 k. The latter is provided in EViews under View/ Residual Diagnostics/Correlogram Q-statistics for residuals, and under View/ Correlogram for the series themselves. 31

32 On the basis of autocovariance functions one can preliminary infer the order of an ARMAprocess Theoretically: ======================================================= acf pacf AR(p) Tails off Cut off after p MA(q) Cut off after q Tails off ARMA(p,q) Tails off Tails off ======================================================= acf = autocorrelation function pacf = partial autocorrelation function Example: (Figure II.5.3 continued) Correlogram of Residuals Date: 01/21/13 Time: 17:17 Sample: 9/20/2005 6/08/2007 Included observations: 400 Autocorrelation Partial Correlation AC PAC Q-Stat Prob The ACF and PACF of the series suggest an ARMA(p, q) model with p no larger than two. 32

33 Example: continued. Neither a simple AR(1) model nor a simple MA(1) model are appropriate, because they have still significant autocorrelation in the residuals left (not shown). However, there is no autocorrelation left in the residuals of an ARMA(1,1) model, and it provides a reasonable fit of the ACF and PACF of y t (available in EViews under View/ARMA Structure.../Correlogram):.6 Autocorrelation Actual Theoretical Partial autocorrelation Actual Theoretical 33

34 Example: continued. Also the residuals or an AR(2) model are white noise (not shown), but the fit to the ACF and PACF of y t is considerably worse:.6 Autocorrelation Actual Theoretical Partial autocorrelation Actual Theoretical 34

35 Example: continued. An ARMA(2,1) has a slightly improved fit to the ACF and PACF of y t as compared to an ARMA(1,1) model (not shown), but the AR(2) term is deemed insignificant: Dependent Variable: SPREAD Method: Least Squares Date: 01/21/13 Time: 18:00 Sample (adjusted): 9/22/2005 6/08/2007 Included observations: 398 after adjustments Convergence achieved after 8 iterations MA Backcast: 9/21/2005 Variable Coefficient Std. Error t-statistic Prob. C AR(1) AR(2) MA(1) 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 Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots Inverted MA Roots.78 We conclude that an ARMA(1,1) model is most appropriate for modelling the spread in Figure II

36 Other popular tools for detecting the order of the model are Akaike s (1974) information criterion (AIC) (72) AIC(p, q) = log ˆσ u 2 + 2(p + q)/t or Schwarz s (1978) Bayesian information criterion (BIC) (75) BIC(p, q) = log(ˆσ 2 ) + (p + q) log(t )/T. There are several other similar criteria, like Hannan and Quinn (HQ). The best fitting model in terms of the chosen criterion is the one that minimizes the criterion. The criteria may end up with different orders of the model! More generally these criteria are of the form (73) AIC(m) = 2l(ˆθ m ) + 2m and (74) BIC(m) = 2l(ˆθ m ) + log(t )m, where ˆθ m is the MLE of θ m, a parameter with m components, l(ˆθ m ) is the value of the log-likelihood at ˆθ m. 36

37 Example: Figure II.5.3 continued. ============================ p q AIC BIC * * ============================ * = minimum The Schwarz criterion suggests ARMA(1,1), whereas AIC suggests ARMA(2,2) due to an even better fit. However, the AR(1) term in the ARMA(2,2) model is deemed insignificant (not shown). Reestimating the model without the AR(1) term yields AIC= and BIC= This is the best model if we prefer goodness of fit over parsimony. ARMA(1,1) is the best model if we prefer parsimony over goodness of fit. 37

38 3.2 Random Walks and Efficient Markets According to the efficient market hypothesis all public information currently available is immediately incorporated into current prices. This means that any new information arriving tomorrow is independent of the price today. A price process with this property is called a random walk. Formally we say that a process is a random walk (RW) if it is of the form (76) y t = µ + y t 1 + u t, where µ is the expected change of the process (drift) and u t i.i.d.(0, σu 2 ). (i.i.d. means independent and identically distributed.) Alternative forms of RW assume only that the u t s are independent (e.g. variances can change) or just that the u t s are uncorrelated (autocorrelations are zero). 38

39 Martingales It can be shown that properly discounted prices in arbitrage-free markets must be martingales, which in turn are defined in terms of so called conditional expectations (77) E t (X) := E(X I t ), that is, expected values conditional upon the information set I t available at time t, which contains the realisations of all random variables known by time t. E t (X) is calculated like an ordinary expectation, however replacing all random variables, the outcomes of which are known by time t, with these outcomes. It may be interpreted as the best forecast we can make for X using all information available at time t. Hence the ordinary (unconditional) expectation E(X) may be thought of as a conditional expectation at time t = : (78) E(X) = E (X). 39

40 A stochastic process y t is called a martingale if the best forecast of future realizations y t+s is the the realization at time t, that is, (79) E t (y t+s ) = y t. The random walk (76) is a special case of a martingale if µ = 0 since then E t (y t+s ) = E t (y t+s 1 + u t+s ) = E t y t + = y t + E t = y t. t+s u t+k k=1 t+s u t+k k=1 The last equality follows from the fact that by independence of the innovations u t no information available before t + k helps in forecasting u t+k, such that E t (u t+k ) = E(u t+k ) = 0. 40

41 The Law of Iterated Expectations Calculations involving conditional expectations make frequently use of the law of iterated expectations, which states that optimal forecasts of random variables cannot be improved upon by producing optimal forecasts of future optimal forecasts, formally: (80) E t (X) = E t E t+s (X) = E t+s E t (X), where s 0, that is, the information set I t+s at time point t+s contains at least the information available at time point t (formally: I t I t+s ). This implies in particular that (81) E(X) = EE t (X) = E t E(X) for all t. Repeated application of (80) yields (82) E t E t+1 E t+2... = E t. 41

42 Example: Rational Expectation Hypothesis Muth formulated in 1961 the rational expectation hypothesis, which says that in the aggregate market participants behave as if they knew the true probability distribution of an assets next periods price and use all available information at time t in order to price the asset such that its current price S t equals the best available forecast of the price one period ahead, S t = E t (S t+1 ), which is just the defining property of a martingale. The forecasting errors ɛ t+1 = S t+1 E t (S t+1 ) are serially uncorrelated with zero mean since by the law of iterated expectations and E(ɛ t+1 ) = EE t (ɛ t+1 ) = E(E t (S t+1 ) E t (S t+1 )) = 0 Cov(ɛ t+1, ɛ t+s ) = E(ɛ t+1 ɛ t+s ) = EE t+s 1 (ɛ t+1 ɛ t+s ) = E(ɛ t+1 E t+s 1 ɛ t+s ) = E(ɛ t+1 0) = 0. Discounted price changes are then also uncorrelated with zero mean since under rational expectations S t+1 S t = S t+1 E t (S t+1 ) = ɛ t+1. 42

43 3.3 Integrated Processes and Stocastic Trends A times series process is said to be integrated of order 1 and denoted I(1) if it is not stationary itself, but becomes stationary after applying the first difference operator (83) := 1 L, that is, y t = y t y t 1 is stationary: (84) y t I(1) y t = µ+y t 1 +u t, u t I(0), where I(0) denotes stationary series. A random walk is I(1) with i.i.d. increments is a special case, but more general I(1)-processes could have autocorrelated differences with moving average components. If a process is I(1), we call it a unit root process and say that it has a stochastic trend. A process is integrated of order d and denoted I(d) if it is not stationary and d is the minimum number of times it must be differenced to achieve stationarity. Integrated processes of order d > 1 are rare. 43

44 ARIMA-model As an example consider the process (85) ϕ(l)y t = θ(l)u t. If, say d, of the roots of the polynomial ϕ(l)=0 are on the unit circle and the rest outside the circle (the process has d unit roots), then ϕ(l) is a nonstationary autoregressive operator. We can write then φ(l)(1 L) d = φ(l) d = ϕ(l) where φ(l) is a stationary autoregressive operator and (86) φ(l) d y t = θ(l)u t which is a stationary ARMA. We say that y t process. follows and ARIMA(p,d,q)- 44

45 A symptom of integrated processes is that the autocorrelations do not tend to die out. Example 3.5: Autocorrelations of Google (log) price series Included observations: 285 =========================================================== Autocorrelation Partial AC AC PAC Q-Stat Prob ===========================================================. *******. ******* ******* ******* * ****** ****** ****** ****** ****** ***** ***** ***** ***** * =========================================================== 45

46 Deterministic Trends The stochastic trend of I(1)-pocesses given by (84) must be distinguished from a deterministic trend, given by (87) y t = α + βt + u t, u t i.i.d(0, σ 2 u ), also called I(0)+trend process. Neither (84) nor (87) is stationary, but the methods to make them stationary differ: I(1)-pocesses must be differenced to become stationary, which is why they are also called difference-stationary. A I(0)+trend process becomes stationary by taking the residuals from fitting a trend line by OLS to the original series, which is why it is also called trend-stationary. Price series are generally difference-stationary, which implies that the proper transformation to render them stationary is taking first differences. 46

47 To see that taking deviations from a trend of a I(1)-pocesses does not make it stationary, iterate (84) to obtain y t = y 0 + µt + t i=1 u i. The term t i=1 u i is clearly non-stationary because its variance increases with t. On the other hand, applying first differences to the trend-stationary process (87) (which should in fact be detrended by taking the residuals of an OLS trend), introduces severe negative autocorrelation: y t y t 1 = (α+βt+u t ) (α+β(t 1) + u t 1 ) = β + u t u t 1, which is an MA(1) process with first-order autocorrelation coefficient

48 Testing for unit roots Consider the general model (88) y t = α + βt + φy t 1 + u t, where u t is stationary. If φ < 1 then the (88) is trend stationary. If φ = 1 then y t is a unit root process (i.e., I(1)) with trend (and drift). Thus, testing whether y t is a unit root process reduces to testing whether φ = 1. 48

49 The ordinary OLS approach does not work! One of the most popular tests is the Augmented Dickey-Fuller (ADF). Other tests are e.g. Phillips-Perron and KPSS-test. Dickey-Fuller regression (89) y t = µ + βt + γy t 1 + u t, where γ = φ 1. The null hypothesis is: y t I(1), i.e., (90) H 0 : γ = 0. This is tested with the usual t-ratio. (91) t = ˆγ s.e.(^γ). 49

50 However, under the null hypothesis (90) the distribution is not the standard t-distribution. Distributions fractiles are tabulated under various assumptions (whether the trend is present (β 0) and/or the drift (α) is present. For example, including a constant but no trend: n : α = 1% α = 5% α = 10% t-statistics which are more negative than the critical values above lead to a rejection of a unit root in favor of a stationary series. In practice also AR-terms are added into the regression to make the residual as white noise as possible. Doing so yields critical values different from the table above. 50

51 Example: Google weekly log prices. 6.8 Google weekly log price III IV I II III IV I II III IV I II III IV I II III IV I II III IV I The series does not appear to be mean-reverting, so we expect it to contain a unit root. 51

### Forecasting the US Dollar / Euro Exchange rate Using ARMA Models

Forecasting the US Dollar / Euro Exchange rate Using ARMA Models LIUWEI (9906360) - 1 - ABSTRACT...3 1. INTRODUCTION...4 2. DATA ANALYSIS...5 2.1 Stationary estimation...5 2.2 Dickey-Fuller Test...6 3.

### Time Series Analysis

Time Series Analysis Identifying possible ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos

### Time Series Analysis 1. Lecture 8: Time Series Analysis. Time Series Analysis MIT 18.S096. Dr. Kempthorne. Fall 2013 MIT 18.S096

Lecture 8: Time Series Analysis MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis 1 Outline Time Series Analysis 1 Time Series Analysis MIT 18.S096 Time Series Analysis 2 A stochastic

### Chapter 12: Time Series Models

Chapter 12: Time Series Models In this chapter: 1. Estimating ad hoc distributed lag & Koyck distributed lag models (UE 12.1.3) 2. Testing for serial correlation in Koyck distributed lag models (UE 12.2.2)

### Univariate Time Series Analysis; ARIMA Models

Econometrics 2 Spring 25 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Outline of the Lecture () Introduction to univariate time series analysis. (2) Stationarity. (3) Characterizing

### Some useful concepts in univariate time series analysis

Some useful concepts in univariate time series analysis Autoregressive moving average models Autocorrelation functions Model Estimation Diagnostic measure Model selection Forecasting Assumptions: 1. Non-seasonal

### Performing Unit Root Tests in EViews. Unit Root Testing

Página 1 de 12 Unit Root Testing The theory behind ARMA estimation is based on stationary time series. A series is said to be (weakly or covariance) stationary if the mean and autocovariances of the series

### Time Series Analysis

Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos (UC3M-UPM)

### 1 Short Introduction to Time Series

ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The

### Analysis and Computation for Finance Time Series - An Introduction

ECMM703 Analysis and Computation for Finance Time Series - An Introduction Alejandra González Harrison 161 Email: mag208@exeter.ac.uk Time Series - An Introduction A time series is a sequence of observations

### Univariate Time Series Analysis; ARIMA Models

Econometrics 2 Fall 25 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Univariate Time Series Analysis We consider a single time series, y,y 2,..., y T. We want to construct simple

### Analysis of Financial Time Series with EViews

Analysis of Financial Time Series with EViews Enrico Foscolo Contents 1 Asset Returns 2 1.1 Empirical Properties of Returns................. 2 2 Heteroskedasticity and Autocorrelation 4 2.1 Testing for

### A Multiplicative Seasonal Box-Jenkins Model to Nigerian Stock Prices

A Multiplicative Seasonal Box-Jenkins Model to Nigerian Stock Prices Ette Harrison Etuk Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Nigeria Email: ettetuk@yahoo.com

### The relationship between stock market parameters and interbank lending market: an empirical evidence

Magomet Yandiev Associate Professor, Department of Economics, Lomonosov Moscow State University mag2097@mail.ru Alexander Pakhalov, PG student, Department of Economics, Lomonosov Moscow State University

### Air passenger departures forecast models A technical note

Ministry of Transport Air passenger departures forecast models A technical note By Haobo Wang Financial, Economic and Statistical Analysis Page 1 of 15 1. Introduction Sine 1999, the Ministry of Business,

### 2. Linear regression with multiple regressors

2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions

### Univariate and Multivariate Methods PEARSON. Addison Wesley

Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston

### ITSM-R Reference Manual

ITSM-R Reference Manual George Weigt June 5, 2015 1 Contents 1 Introduction 3 1.1 Time series analysis in a nutshell............................... 3 1.2 White Noise Variance.....................................

### Financial TIme Series Analysis: Part II

Department of Mathematics and Statistics, University of Vaasa, Finland January 29 February 13, 2015 Feb 14, 2015 1 Univariate linear stochastic models: further topics Unobserved component model Signal

### Lecture 2: ARMA(p,q) models (part 3)

Lecture 2: ARMA(p,q) models (part 3) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC) Univariate time series Sept.

### Sales forecasting # 2

Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univ-rennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting

### Luciano Rispoli Department of Economics, Mathematics and Statistics Birkbeck College (University of London)

Luciano Rispoli Department of Economics, Mathematics and Statistics Birkbeck College (University of London) 1 Forecasting: definition Forecasting is the process of making statements about events whose

### Forecasting Using Eviews 2.0: An Overview

Forecasting Using Eviews 2.0: An Overview Some Preliminaries In what follows it will be useful to distinguish between ex post and ex ante forecasting. In terms of time series modeling, both predict values

### Time Series Analysis

Time Series 1 April 9, 2013 Time Series Analysis This chapter presents an introduction to the branch of statistics known as time series analysis. Often the data we collect in environmental studies is collected

### Chapter 6: Multivariate Cointegration Analysis

Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration

### The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.

Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium

### Non-Stationary Time Series andunitroottests

Econometrics 2 Fall 2005 Non-Stationary Time Series andunitroottests Heino Bohn Nielsen 1of25 Introduction Many economic time series are trending. Important to distinguish between two important cases:

### Time Series Analysis

Time Series Analysis Autoregressive, MA and ARMA processes Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 212 Alonso and García-Martos

### Time Series Analysis

Time Series Analysis Andrea Beccarini Center for Quantitative Economics Winter 2013/2014 Andrea Beccarini (CQE) Time Series Analysis Winter 2013/2014 1 / 143 Introduction Objectives Time series are ubiquitous

### Vector Time Series Model Representations and Analysis with XploRe

0-1 Vector Time Series Model Representations and Analysis with plore Julius Mungo CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin mungo@wiwi.hu-berlin.de plore MulTi Motivation

### Time Series Graphs. Model ACF PACF. White Noise All zeros All zeros. AR(p) Exponential Decay P significant lags before dropping to zero

Time Series Graphs Model ACF PACF White Noise All zeros All zeros AR(p) Exponential Decay P significant lags before dropping to zero MA(q) q significant lags before dropping to zero Exponential Decay ARMA(p,q)

### Eviews Tutorial. File New Workfile. Start observation End observation Annual

APS 425 Professor G. William Schwert Advanced Managerial Data Analysis CS3-110L, 585-275-2470 Fax: 585-461-5475 email: schwert@schwert.ssb.rochester.edu Eviews Tutorial 1. Creating a Workfile: First you

### Price volatility in the silver spot market: An empirical study using Garch applications

Price volatility in the silver spot market: An empirical study using Garch applications ABSTRACT Alan Harper, South University Zhenhu Jin Valparaiso University Raufu Sokunle UBS Investment Bank Manish

### Time Series Analysis

Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Identification of univariate time series models, cont.:

### TIME SERIES ANALYSIS

TIME SERIES ANALYSIS L.M. BHAR AND V.K.SHARMA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi-0 02 lmb@iasri.res.in. Introduction Time series (TS) data refers to observations

### Chapter 9: Univariate Time Series Analysis

Chapter 9: Univariate Time Series Analysis In the last chapter we discussed models with only lags of explanatory variables. These can be misleading if: 1. The dependent variable Y t depends on lags of

### RELATIONSHIP BETWEEN STOCK MARKET VOLATILITY AND EXCHANGE RATE: A STUDY OF KSE

RELATIONSHIP BETWEEN STOCK MARKET VOLATILITY AND EXCHANGE RATE: A STUDY OF KSE Waseem ASLAM Department of Finance and Economics, Foundation University Rawalpindi, Pakistan seem_aslam@yahoo.com Abstract:

### 9th Russian Summer School in Information Retrieval Big Data Analytics with R

9th Russian Summer School in Information Retrieval Big Data Analytics with R Introduction to Time Series with R A. Karakitsiou A. Migdalas Industrial Logistics, ETS Institute Luleå University of Technology

### TIME SERIES ANALYSIS

TIME SERIES ANALYSIS Ramasubramanian V. I.A.S.R.I., Library Avenue, New Delhi- 110 012 ram_stat@yahoo.co.in 1. Introduction A Time Series (TS) is a sequence of observations ordered in time. Mostly these

### The Simple Linear Regression Model: Specification and Estimation

Chapter 3 The Simple Linear Regression Model: Specification and Estimation 3.1 An Economic Model Suppose that we are interested in studying the relationship between household income and expenditure on

### Chapter 5: Bivariate Cointegration Analysis

Chapter 5: Bivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie V. Bivariate Cointegration Analysis...

### 4.3 Moving Average Process MA(q)

66 CHAPTER 4. STATIONARY TS MODELS 4.3 Moving Average Process MA(q) Definition 4.5. {X t } is a moving-average process of order q if X t = Z t + θ 1 Z t 1 +... + θ q Z t q, (4.9) where and θ 1,...,θ q

Chapter 7 Chapter Table of Contents OVERVIEW...193 GETTING STARTED...194 TheThreeStagesofARIMAModeling...194 IdentificationStage...194 Estimation and Diagnostic Checking Stage...... 200 Forecasting Stage...205

### Time Series Analysis in Economics. Klaus Neusser

Time Series Analysis in Economics Klaus Neusser May 26, 2015 Contents I Univariate Time Series Analysis 3 1 Introduction 1 1.1 Some examples.......................... 2 1.2 Formal definitions.........................

### Time Series - ARIMA Models. Instructor: G. William Schwert

APS 425 Fall 25 Time Series : ARIMA Models Instructor: G. William Schwert 585-275-247 schwert@schwert.ssb.rochester.edu Topics Typical time series plot Pattern recognition in auto and partial autocorrelations

### Graphical Tools for Exploring and Analyzing Data From ARIMA Time Series Models

Graphical Tools for Exploring and Analyzing Data From ARIMA Time Series Models William Q. Meeker Department of Statistics Iowa State University Ames, IA 50011 January 13, 2001 Abstract S-plus is a highly

### Lecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay

Lecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay Business cycle plays an important role in economics. In time series analysis, business cycle

### Forecasting Thai Gold Prices

1 Forecasting Thai Gold Prices Pravit Khaemasunun This paper addresses forecasting Thai gold price. Two forecasting models, namely, Multiple-Regression, and Auto-Regressive Integrated Moving Average (ARIMA),

### Rob J Hyndman. Forecasting using. 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1

Rob J Hyndman Forecasting using 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1 Outline 1 Regression with ARIMA errors 2 Example: Japanese cars 3 Using Fourier terms for seasonality 4

### A Trading Strategy Based on the Lead-Lag Relationship of Spot and Futures Prices of the S&P 500

A Trading Strategy Based on the Lead-Lag Relationship of Spot and Futures Prices of the S&P 500 FE8827 Quantitative Trading Strategies 2010/11 Mini-Term 5 Nanyang Technological University Submitted By:

### Chapter 5: Basic Statistics and Hypothesis Testing

Chapter 5: Basic Statistics and Hypothesis Testing In this chapter: 1. Viewing the t-value from an OLS regression (UE 5.2.1) 2. Calculating critical t-values and applying the decision rule (UE 5.2.2) 3.

### The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay Solutions to Final Exam

The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (40 points) Answer briefly the following questions. 1. Describe

### Estimating an ARMA Process

Statistics 910, #12 1 Overview Estimating an ARMA Process 1. Main ideas 2. Fitting autoregressions 3. Fitting with moving average components 4. Standard errors 5. Examples 6. Appendix: Simple estimators

### UNIT ROOT TESTING TO HELP MODEL BUILDING. Lavan Mahadeva & Paul Robinson

Handbooks in Central Banking No. 22 UNIT ROOT TESTING TO HELP MODEL BUILDING Lavan Mahadeva & Paul Robinson Series editors: Andrew Blake & Gill Hammond Issued by the Centre for Central Banking Studies,

### PITFALLS IN TIME SERIES ANALYSIS. Cliff Hurvich Stern School, NYU

PITFALLS IN TIME SERIES ANALYSIS Cliff Hurvich Stern School, NYU The t -Test If x 1,..., x n are independent and identically distributed with mean 0, and n is not too small, then t = x 0 s n has a standard

### In this chapter, we cover some more advanced topics in time series econometrics. In

C h a p t e r Eighteen Advanced Time Series Topics In this chapter, we cover some more advanced topics in time series econometrics. In Chapters 10, 11, and 12, we emphasized in several places that using

### C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)}

C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)} 1. EES 800: Econometrics I Simple linear regression and correlation analysis. Specification and estimation of a regression model. Interpretation of regression

### Booth School of Business, University of Chicago Business 41202, Spring Quarter 2015, Mr. Ruey S. Tsay. Solutions to Midterm

Booth School of Business, University of Chicago Business 41202, Spring Quarter 2015, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has

### Source engine marketing: A preliminary empirical analysis of web search data

Source engine marketing: A preliminary empirical analysis of web search data ABSTRACT Bruce Q. Budd Alfaisal University The purpose of this paper is to empirically investigate a website performance and

### CONSOLIDATED EDISON COMPANY OF NEW YORK, INC. VOLUME FORECASTING MODELS. Variable Coefficient Std. Error t-statistic Prob.

PAGE 1 OF 6 SC 1 (RESIDENTIAL AND RELIGIOUS) Dependent Variable: DLOG(GWH17/BDA0,0,4) Convergence achieved after 16 iterations MA Backcast: 1987Q2 C 0.011618 0.003667 3.168199 0.002100 DLOG(PRICE17S(-3),0,4)

### Time Series Analysis

Time Series Analysis Time series and stochastic processes Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos

### Chapter 5. Analysis of Multiple Time Series. 5.1 Vector Autoregressions

Chapter 5 Analysis of Multiple Time Series Note: The primary references for these notes are chapters 5 and 6 in Enders (2004). An alternative, but more technical treatment can be found in chapters 10-11

### Time Series Analysis

JUNE 2012 Time Series Analysis CONTENT A time series is a chronological sequence of observations on a particular variable. Usually the observations are taken at regular intervals (days, months, years),

### Estimation and Inference in Cointegration Models Economics 582

Estimation and Inference in Cointegration Models Economics 582 Eric Zivot May 17, 2012 Tests for Cointegration Let the ( 1) vector Y be (1). Recall, Y is cointegrated with 0 cointegrating vectors if there

### Time Series Analysis and Forecasting

Time Series Analysis and Forecasting Math 667 Al Nosedal Department of Mathematics Indiana University of Pennsylvania Time Series Analysis and Forecasting p. 1/11 Introduction Many decision-making applications

### Time Series in Mathematical Finance

Instituto Superior Técnico (IST, Portugal) and CEMAT cnunes@math.ist.utl.pt European Summer School in Industrial Mathematics Universidad Carlos III de Madrid July 2013 Outline The objective of this short

### Econometria dei mercati finanziari c.a. A.A. 2011-2012. 4. AR, MA and ARMA Time Series Models. Luca Fanelli. University of Bologna

Econometria dei mercati finanziari c.a. A.A. 2011-2012 4. AR, MA and ARMA Time Series Models Luca Fanelli University of Bologna luca.fanelli@unibo.it For each class of models considered in these slides

### Discrete Time Series Analysis with ARMA Models

Discrete Time Series Analysis with ARMA Models Veronica Sitsofe Ahiati (veronica@aims.ac.za) African Institute for Mathematical Sciences (AIMS) Supervised by Tina Marquardt Munich University of Technology,

### 2. Pooled Cross Sections and Panels. 2.1 Pooled Cross Sections versus Panel Data

2. Pooled Cross Sections and Panels 2.1 Pooled Cross Sections versus Panel Data Pooled Cross Sections are obtained by collecting random samples from a large polulation independently of each other at different

### Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem

Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become

### Is the Forward Exchange Rate a Useful Indicator of the Future Exchange Rate?

Is the Forward Exchange Rate a Useful Indicator of the Future Exchange Rate? Emily Polito, Trinity College In the past two decades, there have been many empirical studies both in support of and opposing

### Modeling and forecasting regional GDP in Sweden. using autoregressive models

MASTER THESIS IN MICRODATA ANALYSIS Modeling and forecasting regional GDP in Sweden using autoregressive models Author: Haonan Zhang Supervisor: Niklas Rudholm 2013 Business Intelligence Program School

### Forecasting of Paddy Production in Sri Lanka: A Time Series Analysis using ARIMA Model

Tropical Agricultural Research Vol. 24 (): 2-3 (22) Forecasting of Paddy Production in Sri Lanka: A Time Series Analysis using ARIMA Model V. Sivapathasundaram * and C. Bogahawatte Postgraduate Institute

### Multiple Linear Regression in Data Mining

Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple

Lecture Notes on Univariate Time Series Analysis and Box Jenkins Forecasting John Frain Economic Analysis, Research and Publications April 1992 (reprinted with revisions) January 1999 Abstract These are

### Determinants of Stock Market Performance in Pakistan

Determinants of Stock Market Performance in Pakistan Mehwish Zafar Sr. Lecturer Bahria University, Karachi campus Abstract Stock market performance, economic and political condition of a country is interrelated

### Chapter 4: Vector Autoregressive Models

Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...

### Econometrics Regression Analysis with Time Series Data

Econometrics Regression Analysis with Time Series Data João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### Global Temperature Trends. Trevor Breusch and Farshid Vahid

ISSN 1440-771X Australia Department of Econometrics and Business Statistics http://www.buseco.monash.edu.au/depts/ebs/pubs/wpapers/ Global Temperature Trends Trevor Breusch and Farshid Vahid March 2011

### Maximum Likelihood Estimation of an ARMA(p,q) Model

Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates

### THE UNIVERSITY OF CHICAGO, Booth School of Business Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay. Solutions to Homework Assignment #2

THE UNIVERSITY OF CHICAGO, Booth School of Business Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay Solutions to Homework Assignment #2 Assignment: 1. Consumer Sentiment of the University of Michigan.

### Study & Development of Short Term Load Forecasting Models Using Stochastic Time Series Analysis

International Journal of Engineering Research and Development e-issn: 2278-067X, p-issn: 2278-800X, www.ijerd.com Volume 9, Issue 11 (February 2014), PP. 31-36 Study & Development of Short Term Load Forecasting

### Sacred Heart University John F. Welch College of Business

Sacred Heart University John F. Welch College of Business CONNECTICUT ECONOMIC OUTLOOK for 2013-2015: A Perspective from Sacred Heart University Students in Business Economics Final Research Project for

### IMPACT OF WORKING CAPITAL MANAGEMENT ON PROFITABILITY

IMPACT OF WORKING CAPITAL MANAGEMENT ON PROFITABILITY Hina Agha, Mba, Mphil Bahria University Karachi Campus, Pakistan Abstract The main purpose of this study is to empirically test the impact of working

### 2. What are the theoretical and practical consequences of autocorrelation?

Lecture 10 Serial Correlation In this lecture, you will learn the following: 1. What is the nature of autocorrelation? 2. What are the theoretical and practical consequences of autocorrelation? 3. Since

### Time Series Analysis

Time Series Analysis Lecture Notes for 475.726 Ross Ihaka Statistics Department University of Auckland April 14, 2005 ii Contents 1 Introduction 1 1.1 Time Series.............................. 1 1.2 Stationarity

### European Journal of Business and Management ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online) Vol.5, No.30, 2013

The Impact of Stock Market Liquidity on Economic Growth in Jordan Shatha Abdul-Khaliq Assistant Professor,AlBlqa Applied University, Jordan * E-mail of the corresponding author: yshatha@gmail.com Abstract

### Vector autoregressions, VAR

1 / 45 Vector autoregressions, VAR Chapter 2 Financial Econometrics Michael Hauser WS15/16 2 / 45 Content Cross-correlations VAR model in standard/reduced form Properties of VAR(1), VAR(p) Structural VAR,

### Introduction to ARMA Models

Statistics 910, #8 1 Overview Introduction to ARMA Models 1. Modeling paradigm 2. Review stationary linear processes 3. ARMA processes 4. Stationarity of ARMA processes 5. Identifiability of ARMA processes

### ADVANCED FORECASTING MODELS USING SAS SOFTWARE

ADVANCED FORECASTING MODELS USING SAS SOFTWARE Girish Kumar Jha IARI, Pusa, New Delhi 110 012 gjha_eco@iari.res.in 1. Transfer Function Model Univariate ARIMA models are useful for analysis and forecasting

### Forecasting model of electricity demand in the Nordic countries. Tone Pedersen

Forecasting model of electricity demand in the Nordic countries Tone Pedersen 3/19/2014 Abstract A model implemented in order to describe the electricity demand on hourly basis for the Nordic countries.

### Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### Forecasting GDP growth

Forecasting GDP growth Torsten Lisson (contact: torsten.lisson@gmail.com Emanuel Gasteiger (contact: emanuel.gasteiger@gmail.com Note: This talk was given at the class Economic Forecasting of Prof. Robert.

### Calculate the holding period return for this investment. It is approximately

1. An investor purchases 100 shares of XYZ at the beginning of the year for \$35. The stock pays a cash dividend of \$3 per share. The price of the stock at the time of the dividend is \$30. The dividend

### CHAPTER 5. Exercise Solutions

CHAPTER 5 Exercise Solutions 91 Chapter 5, Exercise Solutions, Principles of Econometrics, e 9 EXERCISE 5.1 (a) y = 1, x =, x = x * * i x i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y * i (b) (c) yx = 1, x = 16, yx

### Powerful new tools for time series analysis

Christopher F Baum Boston College & DIW August 2007 hristopher F Baum ( Boston College & DIW) NASUG2007 1 / 26 Introduction This presentation discusses two recent developments in time series analysis by