Econometrics II. Seppo Pynnönen. Spring Department of Mathematics and Statistics, University of Vaasa, Finland

Department of Mathematics and Statistics, University of Vaasa, Finland Spring 2018

Part IV Financial Time Series As of Feb 5, 2018

1 Financial Time Series Asset Returns Simple returns Log-returns Portfolio returns: Excess Return Three major Stylized Facts Wold Decomposition Autoregressive Moving Average (ARMA) model Autocorrelation Partial Autocorrelation Estimation of acf Statistical inference ARIMA-model Random Walk Martingale The Law of Iterated Expectations Unit root Testing for unit root

Asset Returns 1 Financial Time Series Asset Returns Simple returns Log-returns Portfolio returns: Excess Return Three major Stylized Facts Wold Decomposition Autoregressive Moving Average (ARMA) model Autocorrelation Partial Autocorrelation Estimation of acf Statistical inference ARIMA-model Random Walk Martingale The Law of Iterated Expectations Unit root Testing for unit root

Asset Returns Simple return: R t = P t + d t P t 1 P t 1, (1) where P t is the price of an asset at time point t and d t is the dividend. Gross return: 1 + R t = P t + d t P t 1. (2) In the following we assume that dividends are included in P t.

Asset Returns Multiperiod gross return: 1 + R t [k] = Pt P t k = P t Pt 1 P t 1 P t 2 P t k+1 P t k = (1 + R t )(1 + R t 1 ) (1 + R t k+1 ) (3) = k 1 j=0 (1 + R t j). Annualized (p.a): Let k denote the return period measured in years, R(p.a) = (1 + R t [k]) 1/k 1 (4) is the simple annualized return.

Asset Returns r t = log(p t /P t 1 ) = log(1 + R t ). (5) k 1 k 1 r t [k] = log(1 + R t j ) = r t j (6) j=0 j=0

Asset Returns Log-returns are called continuously compounded returns. In daily or higher frequency r t R t. Thus, does not make big difference which one is used. Log-returns are preferred in research. Remark 1 Simple returns are multiplicative, log returns are additive. For a discussion, see Levy, et al. (2001) Management Science

Asset Returns Portfolio of n assets with weights w 1,..., w n, w 1 + + w n = 1. Then n R p,t = w i R it, (7) where R p,t is the portfolio return. In terms of log-returns i=1 r p,t n w i r it, (8) i=1 where r p,t is the continuously compounded return of the portfolio.

Asset Returns r e t = r t r 0t, (9) where rt e is the excess return and r 0,t is typically the return of a riskless short-term asset, like three months government bond (loosely bank account ). The riskless return is usually given in annual terms. Thus, it must be scaled to match the time period of the asset return r t. r e t retrun of a zero-investment porfolio.

Three major Stylized Facts 1 Financial Time Series Asset Returns Simple returns Log-returns Portfolio returns: Excess Return Three major Stylized Facts Wold Decomposition Autoregressive Moving Average (ARMA) model Autocorrelation Partial Autocorrelation Estimation of acf Statistical inference ARIMA-model Random Walk Martingale The Law of Iterated Expectations Unit root Testing for unit root

Three major Stylized Facts 1. Return distribution is non-normal - approximately symmetric - fat tails - high peak 2. Almost zero autocorrelation (daily) 3. Autocorrelated squared or absolute value returns

Three major Stylized Facts Example 1 Google s weekly returns from Aug 2004 to Jan 2010 Google's Weekly Returns [2004-2010] -20-10 0 10 2005 2006 2007 2008 2009 2010

Definition 1 Time series y t, t = 1,..., T is covariance stationary if E[y t ] = µ, for all t (10) cov[y t, y t+k ] = γ k, for all t (11) var[y t ] = γ 0 (< ), for all t (12) Series that are not stationary are called nonstationary.

Definition 2 Definition 2: Time series u t is a white noise process if E[u t ] = µ, for all t cov[u t, u s ] = 0, for all t s (13) var[u t ] = σu 2 <, for all t. We denote u t WN(µ, σu). 2 Remark 2 Usually it is assumed in (13) that µ = 0. Remark 3 A WN-process is obviously stationary.

Theorem 2 (Wold Decomposition) Any covariance stationary process y t, t =..., 2, 1, 0, 1, 2,... can be written as y t = µ + u t + a 1 u t 1 +... = µ + a h u t h, (14) h=0 where a 0 = 1 and u t WN(0, σ 2 u), and h=0 a2 h <.

Definition 3 Lag polynomial a(l) = a 0 + a 1 L + a 2 L 2 +, (15) where L is the lag-operator such that Ly t = y t 1. (16) Definition 4 Difference operator y t = y t y t 1. (17)

Thus, in terms of the lag polynomial, equation (14) can be written in short y t = µ + a(l)u t. (18) Note that L k y t = y t k and y t = (1 L)y t.

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 y t = φ 0 + φ 1 y t 1 + + φ p y t p +u t θ 1 u t 1 θ q u t q (19)

In terms of lag-polynomials φ(l) = 1 φ 1 L φ 2 L 2 φ p L p (20) θ(l) = 1 θ 1 L θ 2 L 2 θ q L q (21) the ARMA(p,q) in (19) can be written shortly as φ(l)y t = φ 0 + θ(l)u t (22) or where µ = E[y t ] = y t = µ + θ(l) φ(l) u t, (23) φ 0 1 φ 1 φ p. (24)

If q = 0 the process is called an AR(p)-process and if p = 0 the process is called an MA(q)-process. Example 3 AR(1)-process An AR(1)-process is stationary if φ 1 < 1. y t = φ 0 + φ 1 y t 1 + u t. (25) Below is a sample path for an AR(1)-process with T = 100 observations for φ 0 = 2, φ 1 = 0.7, and u t NID(0, σ 2 u) with σ 2 u = 4 (i.e., standard deviation σ u = 2).

Sample path of an AR(1)-priocess y -5 0 5 10 0 20 40 60 80 100 time

A sample path for an MA(1)-process y t = µ + u t θ 1 u t 1 (26) with µ = 0.67 and θ 1 = 0.7, and u t NID(0, 4). Sample path of an MA(1)-priocess y ma1-5 0 5 0 20 40 60 80 100 time

Autocorvariance Function γ k = cov[y t, y t k ] = E[(y t µ)(y t k µ)] (27) k = 0, 1, 2,.... Variance: γ 0 = var[y t ]. Autocorrelation function ρ k = γ k γ 0. (28) Autocovariances and autocorrelations are symmetric. That is, γ k = γ k and ρ k = ρ k.

For an AR(p)-process the autocorrelation function is of the form ρ k = φ 1 ρ k 1 + φ 2 ρ k 2 + + φ p ρ k p. (29) k > 0. For an MA(q)-process the autocorrelation function is of the form ρ k = θ k + θ 1 θ k 1 + + θ q k θ q 1 + θ 2 1 + + θ2 q (30) for k = 1, 2,..., q and ρ k = 0 for k > q.

Example 4 For an AR(1) process y t = φ 0 + φ 1 y t 1 + u t the autocorrelation function is ρ k = φ k 1. (31) For an MA(1)-process y t = µ + u t θu t 1 the autocorrelation function is ρ k = θ 1 + θ 2, for k = 1 0, for k > 1 (32)

Typically the autocorrelation function is presented graphically by correlogram Autocorrelation function of AR(1) with phi = 0.7 rho 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 lag Autocorrelation function of AR(1) with phi = -0.7 rho -1.0-0.5 0.0 0.5 1.0 1 2 3 4 5 6 lag

An AR(p)-process is stationary if the roots of the polynomial φ(l) = 0 (33) are outside the unit circle (be greater than 1 in absolute value). Alternatively, if we consider the characteristic polynomial m p φ 1 m p 1 φ p = 0, (34) then an AR(p)-process is stationary if the roots of the characteristic polynomial are inside the unit circle (be less than one in absolute value).

An MA-process is always stationary. We say that and MA(q)-process is invertible if the roots of the characteristic polynomial (35) lie outside the unit circle. θ(l) = 1 θ 1 L θ 2 L 2 θ q L q = 0 Invertibility means that an MA-process can be represented as infinite AR-process.

Partial autocorrelation of a time series y t at lag k measures the correlation of y t and y t k after adjusting y t for the effects of y t 1,..., y t k+1. Partial autocorrelations are measured by φ kk which is the last coefficient α k, in regression Thus, denoting y t = φ 0k + φ 1k y t 1 + + φ kk y t k + v t (36) ỹ t = y t (φ 0k + φ 1k y t 1 + + φ k 1,k y t k+1 ) then φ kk = corr[ỹ t, y t k ]. For an AR(p)-process φ kk = 0 for k > p.

Autocorrelation (and partial autocorrelation) functions are estimated by the empirical counterparts ˆγ k = 1 T where is the sample mean. Similarly T k (y t ȳ)(y t k ȳ), (37) t=1 ȳ = 1 T T t=1 y t r k = ˆρ k = ˆγ k ˆγ 0. (38)

It can be shown that if ρ k = 0, then E[r k ] = 0 and asymptotically var[r k ] 1 T. (39) [ ] Similarly, if φ kk = 0 then E ˆφ kk = 0 and asymptotically var[ˆρ kk ] 1 T. (40) In both cases the asymptotic distribution is normal. Thus, testing can be tested with the test statistic H 0 : ρ k = 0, (41) z = T r k, (42) which is asymptotically N(0, 1) distributed under the null hypothesis (41).

Portmanteau statistics to test the hypothesis is due to Box and Pierce (1970) H 0 : ρ 1 = ρ 2 = = ρ m = 0 (43) Q (m) = T m rk 2, (44) k=1 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 Q(m) = T (T + 2) m k=1 1 T k r 2 k. (45)

On the basis of autocovariance function one can preliminary infer the order of an ARMA-proces 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

Other popular tools for detecting the order of the model are Akaike s (1974) information criterion (AIC) AIC(p, q) = log ˆσ 2 u + 2(p + q)/t (46) or Schwarz s (1978) Bayesian information criterion (BIC) 2 BIC(p, q) = log(ˆσ 2 ) + (p + q) log(t )/T. (49) There are several other similar criteria, like Hannan and Quinn (HQ). and 2 More generally these criteria are of the form AIC(m) = 2l(ˆθ m) + 2m (47) BIC(m) = 2l(ˆθ m) + log(t )m, (48) 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.

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!

Example 5 Google weekly (adjusted) closing prices Aug 2004 Jan 2017. Google prices Google weekly returns Price 200 400 600 800 Observed EWMA(0.05) Return (% per week) 10 0 10 20 2006 2010 2014 Time 2006 2010 2014 Time Google retuns Google return autocorrelations Density 0.00 0.04 0.08 0.12 Normal Empirical ACF 0.2 0.1 0.0 0.1 0.2 10 0 10 20 Return 5 10 15 20 25 30 35 Lag

Autocorrelations (AC) and parial autocorrelations (PAC) Google s weekly returns 2004-2017 Included observations: 650 ================================================ lag AC PAC Q-AC Q-PAC p(q-ac) p(q-pac) ------------------------------------------------ 1-0.043-0.043 1.198 1.198 0.274 0.274 2 0.071 0.069 4.478 4.322 0.107 0.115 3 0.016 0.022 4.653 4.647 0.199 0.200 4 0.037 0.034 5.557 5.407 0.235 0.248 5-0.018-0.018 5.767 5.614 0.330 0.346 6 0.056 0.049 7.801 7.209 0.253 0.302 7-0.045-0.040 9.119 8.255 0.244 0.311 8 0.011 0.000 9.196 8.255 0.326 0.409 9-0.001 0.004 9.197 8.265 0.419 0.508 10-0.031-0.034 9.817 9.012 0.457 0.531 ================================================ All autocorrelation and partial autocorrelation estimate virtually to zero and none of the Q-statistics are significant.

============================ p q AIC BIC ---------------------------- 0 0 13107.33* 13119.43* 1 0 13108.98 13127.13 2 0 13110.93 13135.13 0 1 13108.98 13127.13 0 2 13110.94 13135.14 1 1 13110.99 13135.19 2 1 13112.31 13142.56 ============================ * = minimum AIC BIC suggest also white noise.

Later we will find that autocorrelations of the squared returns will be highly significant, suggesting that there is still left time dependency into the series. The dependency, however, is nonlinear by nature.

Consider the process ϕ(l)y t = θ(l)u t. (50) If, say d, of the roots of the polynomial ϕ(l) = 0 are on the unit circle and the rest outside the circle, 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 which is a stationary ARMA. φ(l) d y t = θ(l)u t (51) We say that y t follows and ARIMA(p,d,q)-process. A symptom of unit roots is that the autocorrelations do not tend to die out.

Example 6 Example 5: Autocorrelations of Google (log) price series Included observations: 285 =========================================================== Autocorrelation Partial AC AC PAC Q-Stat Prob ===========================================================. *******. ******* 1 0.972 0.972 272.32 0.000. *******.. 2 0.944-0.020 530.12 0.000. ******* *. 3 0.912-0.098 771.35 0.000. ******.. 4 0.880-0.007 996.74 0.000. ******.. 5 0.850 0.022 1207.7 0.000. ******.. 6 0.819-0.023 1404.4 0.000. ******.. 7 0.790 0.005 1588.2 0.000. ******.. 8 0.763 0.013 1760.0 0.000. *****.. 9 0.737 0.010 1920.9 0.000. *****.. 10 0.716 0.072 2073.4 0.000. *****.. 11 0.698 0.040 2218.7 0.000. ***** *. 12 0.676-0.088 2355.7 0.000 ===========================================================

We say that a process is a random walk (RW) if it is of the form y t = µ + y t 1 + u t, (52) where µ is the expected change of the process (drift) series and u t i.i.d(0, σ 2 u). More general forms of RW assume that u t is independent process (variances can change) or just that u t is uncorrelated process (autocorrelations are zero). Earlier random walk was considered as a useful model for share prices.

Martingale 1 Financial Time Series Asset Returns Simple returns Log-returns Portfolio returns: Excess Return Three major Stylized Facts Wold Decomposition Autoregressive Moving Average (ARMA) model Autocorrelation Partial Autocorrelation Estimation of acf Statistical inference ARIMA-model Random Walk Martingale The Law of Iterated Expectations Unit root Testing for unit root

Martingale A stochastic process is called a martingale with respect to information I t available at time point t if for all t s E[y s I t ] = y t. (53) That is, given information at time point t the best prediction for a future value y s of the stochastic process is the last observed value y t. It is assumed that y t I s for all t s. Martingale is considered as a useful model for the so called fair game, in which the odds of winning (or loosing) for all participants are the same. Martingales constitute the basis for derivative pricing.

Martingale Remark 4 Remark 4: For short the conditional expectation of the form in (53) is usually denoted as E t [y s ] E[y s I t ]. (54)

Martingale For any I t I s, where t s and for any random variable y E t [E s [y]] = E t [y]. (55) Also E s [E t [y]] = E t [y]. (56)

Unit root 1 Financial Time Series Asset Returns Simple returns Log-returns Portfolio returns: Excess Return Three major Stylized Facts Wold Decomposition Autoregressive Moving Average (ARMA) model Autocorrelation Partial Autocorrelation Estimation of acf Statistical inference ARIMA-model Random Walk Martingale The Law of Iterated Expectations Unit root Testing for unit root

Unit root Definition 5 Times series y t is said to be integrated of order 1, if it is of the form denoted as y t I (1), where (1 L)y t = δ + ψ(l)u t, (57) ψ(l) = 1 + ψ 1 L + ψ 2 L 2 + ψ 3 L 3 + (58) such that j=1 ψ j <, ψ(1) 0, roots of ψ(z) = 0 are outside the unit circle [or the polynomial (58) is of order zero], and u t is a white noise series with mean zero and variance σ 2 u.

Unit root Remark 5 If a time series process is of the form of the right hand side of (57), i.e., x t = δ + ψ(l)u t, (59) where ψ(l) satisfies the conditions of Def 5, it can be shown that x t is stationary. In such a case we denote x t I (0), i.e, integrated of order zero. Accordingly a stationary process is an I (0) process.

Unit root Remark 6 The assumption ψ(1) 0 is important. It rules out for example the trend stationary series y t = α + βt + ψ(l)u t. (60) Because E[y t ] = α + βt, y t is nonstationary. However, (1 L)y t = β + ψ(l)u t, (61) where ψ(l) = (1 L)ψ(L). (62) Now, although, (1 L)y t is stationary, however, ψ(1) = (1 1)ψ(1) = 0, which does not satisfy the rule in Definition 5, and hence a trend stationary series is not I (1).

Unit root Consider the general model where u t is stationary. y t = α + βt + φy t 1 + u t, (63) If φ < 1 then the (63) is trend stationary. If φ = 1 then y t is 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.

Unit root 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 where γ = φ 1. y t = µ + βt + γy t 1 + u t, (64) The null hypothesis is: y t I (1), i.e., This is tested with the usual t-ratio. H 0 : γ = 0. (65) t = ˆγ s.e.(^γ). (66)

Unit root However, under the null hypothesis (65) 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. In practice also AR-terms are added into the regression to make the residual as white noise as possible.

Unit root Elliot, Rosenberg and Stock (1996) Econometrica 64, 813 836, propose a modified version of ADF, where the series is first de-trended before applying ADF by GLS estimated trend. In Stata test results are produced at different lags in AR-terms.

Unit root Example 7 Unit root in Google weekly prices Google (adjusted) closing prices Aug 2004 Jan 2017 Price 200 400 600 800 2006 2008 2010 2012 2014 2016 Time

Unit root ===================== (a) No drift no trend ===================== df1 <- ur.df(y = log(gw$aclose), lags = 10, select = "AIC") # by default no drift, no trend excluded Value of test-statistic is: 2.0172 Critical values for test statistics: 1pct 5pct 10pct tau1-2.58-1.95-1.62 =================== (b) Drift, no trend =================== df2 <- ur.df(y = log(gw$aclose), type = "drift", lags = 10, select = "AIC") Value of test-statistic is: -1.4021 3.3159 Critical values for test statistics: 1pct 5pct 10pct tau2-3.43-2.86-2.57 phi1 6.43 4.59 3.78 =================== (c) Drift and trend =================== df3 <- ur.df(y = log(gw$aclose), type = "trend", lags = 10, select = "AIC") Value of test-statistic is: -2.7589 3.9588 3.8746 Critical values for test statistics: 1pct 5pct 10pct tau3-3.96-3.41-3.12 phi2 6.09 4.68 4.03 phi3 8.27 6.25 5.34 # DF with drift # DF with drift and trend The null hypothesis of unit root is not rejected.

Unit root I(2)? Trend is not needed in ADF here. ====================== (a) No drift, no trend ====================== Value of test-statistic is: -16.9747 Critical values for test statistics: 1pct 5pct 10pct tau1-2.58-1.95-1.62 ========================= (b) Drift, no trend ========================= Value of test-statistic is: -17.1464 146.9996 Critical values for test statistics: 1pct 5pct 10pct tau2-3.43-2.86-2.57 phi1 6.43 4.59 3.78 The unit root in log price changes, i.e., returns, is clearly rejected. The graph below of supports the stationarity of the return (differences of log price) series.

Unit root Google's weekly log returns Aug 2004 Jan 2017 Return 10 0 10 20 2006 2008 2010 2012 2014 2016 Time Thus, we can conclude that log(close) I (1).

Unit root DF-GLS leads to the same conclusion. ==================================== (a) ERS with constant, no trend ==================================== Value of test-statistic is: 1.2058 Critical values of DF-GLS are: 1pct 5pct 10pct critical values -2.57-1.94-1.62 ================================= (b) ERS with trend ================================ Value of test-statistic is: -1.2833 Critical values of DF-GLS are: 1pct 5pct 10pct critical values -3.48-2.89-2.57 The overall conclusion is that Google s price series is difference stationary, i.e., I (1), and thus not trend stationary.