1 A Primer on Financial Time Series Analysis Elements of Financial Risk Management Chapter Peter Christoffersen Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Overview Topics under discussion in this Chapter •Probability Distributions and Moments •The Linear Model •Univariate Time Series Models •Multivariate Time Series Models Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Common pitfalls encountered while dealing with time series data • Spurious detection of mean-reversion • Spurious regression • Spurious detection of causality Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Univariate Probability Distributions • Let denote the cumulative probability distribution of the random variable • The probability of being less than is given by • Let be the probability density of and assume that is defined from to • Then the probability of having a value of less than Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Univariate Probability Distributions • Therefore, we have • We will also have • The probability of obtaining a value in an interval between and where is Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Univariate Probability Distributions • The expected value or mean of is defined as • Further we can manipulate expectations by Where and are constants Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Univariate Probability Distributions • Variance is a measure of the expected variation of variable around its mean and is defined as, • It can also be written as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Univariate Probability Distributions • We can further write • From this we can construct a new r.v , and if the mean and variance of are zero and one correspondingly then we have, • This proves very useful in constructing random variables with desired mean and variance Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Univariate Probability Distributions • Mean and variance are the first two central moments Third and fourth central moments, also known as skewness and kurtosis are defined by, • Looking closely at the formulas we will see that, Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Univariate Probability Distributions • As an example we can consider the normal distribution with parameters, and • It is defined by • The normal distribution moments are: Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 10 Time Series Regression • If ACF dies off only very slowly, then we need to firstdifference each series and run the regression • Now use the ACF on the residuals of the new regression and check for ACF dynamics • The AR, MA, or ARMA models can be used to model any dynamics in et • After modeling and estimating the parameters in the residual time series, et, the entire regression model including a and b can be reestimated using MLE Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 63 Pitfall #2: Spurious Regression • Consider two completely unrelated times series—each with a unit root • They are likely to appear related in a regression that has a significant b coefficient • Let s1t and s2t be two independent random walks • where are independent of each other and independent over time • True value of b is zero in the time series regression Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 64 Pitfall #2: Spurious Regression 65 • However standard t-tests will tend to conclude that b is nonzero when in truth it is zero • This problem is known as spurious regression • So, use ACF to detect spurious regression • If the relationship between s1t and s2t is spurious then the error term, et; will have a highly persistent ACF and the regression in first differences will not show a significant estimate of b Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen Cointegration Relationships between variables with unit roots are not always spurious • A variable with a unit root is also called integrated • If two variables that are both integrated have a linear combination with no unit root then we say they are cointegrated • Examples: long-run consumption and production in an economy • The spot and the futures price of an asset that are related via a no-arbitrage condition Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 66 Cointegration • The pairs trading strategy consists of two stocks whose prices tend to move together • If prices diverge then we buy the temporarily cheap stock and short sell the temporarily expensive stock and wait for the typical relationship between the prices to return • Such a strategy hinges on the stock prices being cointegrated • Consider a simple bivariate model where Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen 67 Cointegration Note that s1t has a unit root and that the level of s1t and s2t are related via b • Assume that are independent of each other and independent over time • The cointegration model can be used to preserve the relationship between the variables in the longterm forecasts Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 68 Cross-Correlations • Consider two financial time series, R1,t and R2,t • They can be dependent in three possible ways: • can lead (e.g., ) • can lag (e.g., ), • They can be contemporaneously related (e.g., ) • We use cross-correlation matrices to detect all these possible dynamic relationships • The sample cross-correlation matrices are the multivariate analogues of the ACF function Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen 69 Cross-Correlations For a bivariate time series, the cross-covariance matrix for lag is • The two diagonal terms are the autocovariance function of R1,t, and R2,t, respectively • In the general case of a k-dimensional time series, we have • where Rt is now a k by vector of variables Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 70 Cross-Correlations 71 • Detecting lead and lag effects is important • For example, when relating an illiquid stock to a liquid market factor • The illiquidity of the stock implies price observations that are often stale, which in turn will have a spuriously low correlation with the liquid market factor • The stale equity price will be correlated with the lagged market factor and this lagged relationship is used to compute a liquidity-corrected measure of the dependence between the stock and the market Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen Vector Autoregressions (VAR) Consider a first-order Vector Autoregression, call it VAR(1) • where Rt is again a k by vector of variables • The bivariate case is simply Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 72 Vector Autoregressions (VAR) • In the VAR, R1,t and R2,t are related via their covariance • The VAR only depends on lagged variables so, it is immediately useful in forecasting • If the variables included on the right-hand-side of each equation in the VAR are the same then the VAR is called unrestricted • If so, OLS can be used equation-by-equation to estimate the parameters Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 73 Pitfall #3: Spurious Causality • We may want to see if the lagged value of , namely , is causal for the current value of • If so, can be used in forecasting • Consider a simple regression of the form • This regression may easily lead to false conclusions if is persistent and so depends on its own past value Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 74 Pitfall #3: Spurious Causality • To truly assess if causes we need to check if past was useful for forecasting current once the past has been accounted for • This question can be answered by running a VAR model: Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 75 Pitfall #3: Spurious Causality • Now we can define Granger causality as follows: • In some cases several lags of may be needed on the right-hand side of the equation for • We may need more lags of in the equation for Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 76 Summary 77 • Financial asset prices and portfolio values can be viewed as examples of very persistent time series • The three most important issues are • Spurious detection of mean reversion-erroneously finding that a variable is mean-reverting when it is truly a random walk • Spurious regression-erroneously finding that a variable x is significant when regressing y on x • Spurious detection of causality-erroneously finding that the current value of x causes future values of y when in reality it cannot Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen ... Regression Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Figure 3.1 Scatter Plot of Anscombes Four Data Sets with Regression Lines Elements of Financial Risk Management. .. similar fashion by, Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 18 The Linear Model • Linear models of the type below is often used by risk managers, • Where... probability of being less than is given by • Let be the probability density of and assume that is defined from to • Then the probability of having a value of less than Elements of Financial Risk Management