Empirical Market Microstructure Analysis (EMMA)

Transcription

1 Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein Albert-Ludwigs-University of Freiburg Summer Term 2016 Prof. Stein EMMA Lecture 3 Summer Term / 34

2 Outline 1. Introduction: Financial Markets and Market Structure 2. Financial Market Equilibrium Theory and Asset Pricing Models 3. Statistical Building Blocks and Econometric Basics 4. Transaction and Trading Models 5. Information-Based Models 6. Inventory Models 7. Limit Order Book Models 8. Price Discovery and Liquidity 9. High-frequency Trading 10. Current Developments Prof. Stein EMMA Lecture 3 Summer Term / 34

3 Important Notes Having discussed the different market types, trading systems and order types in the first lecture, and the relevant economic equilibrium models in the second lecture, this set of slides concludes the basics part of the course. Here the basic regression model is briefly reviewed before the time-series properties of financial market variables are discussed. The basics of time-series properties will help in understanding some of the models that will be discussed throughout the course. Of course all relevant elements will be discussed in detail when the papers are discussed, the slides here should just serve as a helping file. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

4 Linear regression review The simple regression function with several descriptive variables: y i = α + β 1 x 1i β k x ki + u i with y i = dependent variable α= constant x ki = explanatory variable β k = coefficient of explanatory variable k u i = random error term, independently, identically distributed u i.i.d. The method of least squares minimizes the sum of the squares of the residuals. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

5 Linear regression review min N ûi 2 = (y i ŷ i ) 2 i=1 y i = ˆα + K ˆβ k x ki + û i k=1 ŷ i = ˆα + K ˆβ k x ki k=1 û i = y i ˆα + K ˆβ k x ki û i = y i ŷ i k=1 Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

6 Linear regression review Residual and estimator in case of one variable: N i i=1û 2 = (y i ŷ i ) 2 ûi 2 = (y i ˆα ˆβ 1 x i ) 2 i i N ˆβ i = [(x i x)(y i ȳ)] i=1 N ˆα = y ˆβ 1 x [(x i x)] 2 i=1 Total, explained and residual sum of squares: (y i y) 2 = (ŷ i y) 2 + ûi 2 = (ŷ i y) 2 + (y i ŷ i ) 2 i i i i TSS = ESS + RSS = ESS + RSS Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

7 Linear regression review Source: Brooks (2008) Prof. Stein EMMA Lecture 3 Summer Term / 34

8 Linear regression review Quality of the regression: The explanatory power of the estimate can be represented by several different measures: R 2 = TSS ESS = 1 TSS RSS = 1 û2 i (ŷi y) 2 R 2 = 1 RSS (N K 1) TSS (N 1) R 2 = 1 (1 R 2 ) (N 1) (N K) loglikelihood = N 2 (1 + log(2π) + log( û2 i N ) AIC = 2l N + 2K N SIC = 2l N + K log(n) N Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

9 Hypothesis tests t-test of the null hypothesis that the respective observed coefficient is zero: t k = β k ˆσ(β k ) F-test to test the null hypothesis that all coefficients are zero: F = R 2 K 1 (1 R 2 ) (N K) Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

10 Hypothesis tests Source: Brooks (2008) Prof. Stein EMMA Lecture 3 Summer Term / 34

11 Hypothesis test Source: Brooks (2008) Prof. Stein EMMA Lecture 3 Summer Term / 34

12 Hypothesis test Source: Brooks (2008) Prof. Stein EMMA Lecture 3 Summer Term / 34

13 OLS and MLE Ordinary last squares is one possibility to determine the parameters. Maximum likelihood estimation can be employed as well. Here the log likelihood function for the most simple case is shown: log L = T 2 log(2π) T 2 log(σ2 u) T [ ] (yi α β 1 x 1i ) (other common notations to be found as well) t=1 2σ 2 u Assumed again is the normal distribution of residuals. MLE is done by maximizing the likelihood by choice of the respective parameters. As we will see, a lot of studies that are discussed in the course employ MLE estimations. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

14 Time series properties Studies of financial market data are often operated with time series. Econometric models which consider one or more variables over time are called dynamic models. Generally, a stochastic process is to be described, which is based on the variables under consideration. This process is also called data generating process (DGP). When the data generating process is known, we are able to describe it recursively. This lecture includes the basics of time series analysis, what helps to understand the models discussed later in the course. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

15 Time series properties Modified example of Enders ( Applied Econometric Time Series,2010): A variable is for example consisting of trend, seasonal and irregular part: T t = t S t = 1.6sin ( ) t π 10 I t = 0.7I t 1 + ɛ t 6 Components of the time series 7 Realizations of the time series Source: Own example, modification based on Enders, W. (2010). Applied Econometric Times Series. Wiley. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

16 Stationarity The example above shows that the time series depends on previous observations in all components. For stationarity, different characteristics of these dependencies are essential: A stationary variable is displaying the characteristic of mean reversion, that is, fluctuates around a constant long-term mean (E (Y t ) = µ). Furthermore, stationary time-series possess a finite, constant variance (Var (Y t ) = σ 2 for all t) and are covariance stationary, meaning that the covariance between past observations depends only on the time span between the observations, and not on the time of observation (Cov (Y t,y t k ) = γ k for all k). Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

17 Autoregression Usually one speaks of stationarity if the time series is covariance stationary. This can can be investigated by using the autoregression function: p y t = a 0 + a i y t i + ɛ t i=1 y t = a 0 + a 1 y t 1 + a 2 y t a p y t p + ɛ t The sequence of the error term is a white noise process (the error term is a pure random variable, independently, identically distributed with mean 0 and constant variance) and the linear coefficients are also constant. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

18 Autoregression Using the lag operator the process becomes: ( 1 a1 L 1 + a 2 L a p L p) y t = a 0 + ɛ t a (L)y t = a 0 + ɛ t, with 1 a 1 L + a 2 L a p L p. The lag operator is linear and has a negative sign for future observations. Lag-operator: L i y t = y t i and L i y t = y t+i as well as y t = y t y t 1 = (1 L)y t Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

19 Autoregression As outlined above, the autoregression function: y t = a 0 + a 1 y t 1 + a 2 y t a p y t p + ɛ t Conditions for stationarity of a process derive from the solution of the changed AR-process, ex constant and error terms: y t a 1 L 1 y t a 2 L 2 y t... a p L p y t p = 0 y t ( 1 a1 z a 2 z 2... a p z p) = 0 Here the lag operator is replaced with the complex variable z. The expression in brackets is the characteristic polynomial of the process, and the solutions are the characteristic roots of the AR(p) process. 1 a 1 z = 0 a 1 z = 1 z = 1 a 1 a 1 = 1 z Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

20 Stationarity and Autoregression Is the condition fulfilled, we call the process weakly stationary. For an AR(2) process, the conditions are: (a 1 + a 2 ) < 1 (a 2 a 1 ) < 1 1 < a 2 < 1 A stationary variable is displaying the characteristic of mean reversion, is fluctuating around a constant long-term mean. Furthermore, stationary time-series possess a finite, constant variance and are covariance stationary, meaning that the covariance between past observations depends only on the time span between the observations, and not on the time of observation: E (y t ) = µ Var (y t ) = E (y t µ) 2 = σ 2 Cov (y t,y t s ) = E (y t µ)(y t s µ) = γ s Whereas for all s: µ = const. σ 2 = const. γ s = const. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

21 Integration Definition of integration (Engle and Granger, 1987, Co-Integration and Error-Correction: Representation, Estimation and Testing ): A series with no deterministic trend components which has a stationary, invertible, ARMA representation after differencing d times, is said to be integrated of order d, denoted I(d). This means that a variable which is stationary after differencing once, is integrated of order 1, namely I(1). Testing for stationarity can be accomplished with many different procedures, the most common is the Augmented-Dickey-Fuller-Test (ADF-Test). It is tested whether the process exhibits a unit root, i.e. whether previous observations have a persistent, non-decaying influence on following observations. If the null hypothesis of a unit root can be rejected, the variable is stationary. If not, the variable is non-stationary. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

22 White Noise A white noise process is a pure random process. The observations are independently, identically distributed with a constant mean and constant variance. Additionally the covariance for a lag > 0 is 0: E(y t ) = µ var(y t ) = σ 2 γ t r = { σ2 if t = r 0 otherwise Every observation is independent of previous ones. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

23 Moving Average The sequence from the white noise process is termed moving average function of order q, MA(q): q y t = µ + u t + θ 1 u t θ q u t q = µ + θ i u t i + u t An MA-process is a linear combination of white-noise-processes. A variable y t here is dependent on previous error terms. i=1 Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

24 Moving Average Using the lag operator yields: q y t = µ + θ i L i u t + u t with the exponent of the lag operator giving i=1 the lag order: L i u t = u t i Another transformation yields: y t = µ + θ(l)u t, with: θ(l) = (1 + θ 1 L + θ 2 L θ q L q ) Properties of an MA-process: E(y t ) = µ var(y t ) = γ 0 = (1 + θ θ θ2 q)σ 2 γ s = { (θ s + θ s+1 θ 1 + θ s+2 θ θ q θ q s ) for s = 1,2,,q 0 for s > q Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

25 AR-Process In an autoregressive process the actual value of a variable is dependent on its own value of previous periods and an error term. An AR(p) process can be formulated as: y t = µ + φ 1 y t 1 + φ 2 y t φ p y t p + u t = µ + Using the lag-operator yields: p y t = µ + φ i L i y t + u t i=1 p φ i y t i + u t φ(l)y t = µ + u t with φ(l) = (1 φ 1 L φ 2 L 2 θ p L p ) i=1 Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

26 ARMA An ARMA process is a combination of an AR(p) and an MA(q) process: In an ARMA model the value of a variable is dependent on its own previous realizations and a combination of the current and previous error terms: φ(l)y t = µ + θ(l)u t, with φ(l) = (1 φ 1 L φ 2 L 2 θ p L p ) θ(l) = (1 + θ 1 L + θ 2 L θ q L q ) E(u t ) = 0; E(u 2 t ) = σ 2 ; E(u t u s ) = 0, t s Investigations of ARMA processes can be done systematically for example by using the Box-Jenkins method. Here the whole investigation is fragmented into identification, estimation and evaluation. In practice, ARMA model investigations are often done iteratively however and often need to rely on information criteria. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

27 ACF For a covariance stationary process the autocorrelation function is: T 1 T s (y t ȳ)(y t s ȳ) t=s+1 T ρ s =, with ȳ = T T 1 y t as arithmetic mean 1 (y t ȳ) 2 t=1 T 1 t=s+1 As the autocovariance is constant for covariance stationary processes, so is the autocorrelation. Thus, the autocorrelation is independent of the time. In the case of all autocorrelations being zero, we obtain the white noise process again. The latter is an assumption for the residuals in ordinary least squares, regression analysis. For the case that at least for one time span there is either positive or negative correlation between the residuals, we have autocorrelation in the residuals. This is called serial correlation. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

28 PACF The partial autocorrelation function is expressing the correlation between 2 observations of different time points, where the influence of observations in between is excluded: φ 11 = ρ 1 φ 22 = (ρ 2 ρ 2 1) (1 ρ 2 1) φ ss = s 1 ρ s φ s 1,j ρ s j j=1 s 1 1 j=1 φ s 1,j ρ s j,s = 3,4,5..., with φ sj = φ s 1,j φ ss φ s 1,s j for j = 1,2,...,s 1 Accordingly, the autocorrelation function is informative on the unconditional correlation, whereas the partial autocorrelation function is informative on the conditional correlation between two time points. Both functions are used to identify ARMA structures. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

29 ACF and PACF Source: First table: Enders (2010) // Second table: Rachev et al. (2007) Prof. Stein EMMA Lecture 3 Summer Term / 34

30 AR(I)MA Considering the ARMA specification as difference equation, we can solve it obtaining the so-called moving average representation for the dependent variable: p q y = µ + φ i y t i + θ i u t i i=1 i=1 q µ+ θ p q i u t i (1 φ i L i i=1 )y i = µ + θ i u t i y t = p i=1 i=1 (1 φ i L i ) i=1 The resulting stochastic difference equation can be seen as an MA process of infinite order: q y t = µ+ θ i u t i i=1 p (1 φ i L i ) i=1 MA( ) Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

31 AR(I)MA The main interest in this representation is the condition for stability of the stochastic difference equation being the necessity of convergence of the MA process. It can be shown that this is given for the case that all characteristic roots of the polynomial p (1 φ i L i ) are outside of the unit circle. i=1 In addition, if the dependent variable is explained by a linear stochastic difference equation, the condition for stability is a necessary condition for stationarity of the dependent variable. If all characteristic roots lie outside the unit circle, there is a stable, stationary process. However, if there is at least one root within the unit circle, the sequence is of the endogenous variable is integrated. This results in the presence of an Autoregressive Integrated Moving Average (ARIMA) process. Prof. Stein (michael.stein@vwl.uni-freiburg.de) EMMA Lecture 3 Summer Term / 34

32 Process comparison Source: Brooks(2008) Prof. Stein EMMA Lecture 3 Summer Term / 34

33 Process comparison Source: Brooks(2008) Prof. Stein EMMA Lecture 3 Summer Term / 34

34 References In general, recommended literature for econometrics is the one that was listed in Lecture 1. Prof. Stein EMMA Lecture 3 Summer Term / 34