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Module 17: Time Series Analysis

Module 17 Objective

Analyze univariate and multivariate financial and insurance data (incl. asset prices, credit spreads and defaults, interest rates and insurance losses) using appropriate statistical methods

Recommend a specific choice of model based on the results of both quantitative and qualitative analysis of financial or insurance data

Discuss the fitting of distributions to a given set of data and analysis of time series processes

Use of the techniques here to model specific types of risk will be covered in Part 4

Similar focus and note as the previous section

Definitions

Strict Stationarity

A process \(\{X_t : t= 1,2,...,T\}\) is said to be strictly stationary if:

  • The joint distribution:

    \(X_r,X_{r+1},...,X_s\) and

    \(X_{r+k}, X_{r+k+1},...,X_{s+k}\)

    are identical for all \(r, s, k \in \mathbb{Z}\)

Weak and Covariance Stationarity

A process \(\{X_t : t= 0,1,2,...,T\}\) is said to be covariance stationary (or weakly stationary) if:

  1. The mean of the process \(m(T) = \mathrm{E}(X_t) = \mu\) is constant, and

  2. The covariance of the process:

    \(\mathrm{Cov}(X_r, X_{r+k}) = \mathrm{E}\left[\left(X_r - m(r)\right)\left(X_{r+k} - m(r+k)\right)\right] = \gamma_k\)

    Covariance depends only on the time different (lag) \(k\) between the terms

Weakly stationary of order n \(\Rightarrow\) the moments of subsets of the process are defined and equal up to the nth moment

Importance of stationarity in modeling financial time series

  • Stationarity means that the statistical properties of the process (incl. the moments and the relationships between observations in different periods) stay the same over time

  • Important as it determines the extend to which we can use past data to try to model the future

  • Strict stationarity is very restrictive and is unlikely true in real world data

  • The requirement can be relaxed and we can still analyze the data under weak stationarity

Finite variance process strict stationarity \(\Rightarrow\) Weak stationarity

Infinite variance process strict stationarity \(\Rightarrow \!\!\!\!\!\! \big/\) Weak stationarity

White Noise

A process \(\{\epsilon_t : t = 0,1,2,...,T\}\) is a white noise process if it is:

  1. Covariance stationary

  2. \(\forall\) \(t \in \mathbb{Z}\):

    1. \(\mathrm{Corr}(\epsilon_t,\epsilon_{t+k}) = \begin{cases} 1 & \text{if } k = 0 \\ 0 & \text{if } k \neq 0 \\ \end{cases}\)

    2. \(m(t) = \mathrm{E}(\epsilon_t) = 0\)

White noise process

  • Every element is uncorrelated with any previous observations and the process oscillates randomly around zero

  • Important concept for modeling financial time series as many financial processes have a random element that cannot be known in advance (which we call white noise)

Strict White Noise

A white noise process \(\{\epsilon_t : t = 1,2,...,T\}\) is a strict white noise process if it is:

  1. A set of iid r.v. with finite variance

  2. (Typically) assume \(N(0,\sigma^2)\)

Trend and Difference Stationarity

Trend Stationary

  • Where the observations oscillate randomly around a steadily changing value which is a function of time only

    e.g. \(X_t = \alpha_0 + \alpha_1 t + \epsilon_t\)

Difference Stationarity and Integrated Processes

  • We can difference successive observations to generate a new process which may be stationary (if the observed TS is not) and model the new process

Integrated process of order \(d\) (\(I(d)\) process)

  • One where the process needs to be differenced \(d\) times before the result \(\Delta^d X_t\) is covariance stationary

    • (Difference stationary process) Integrated process of order 1 (\(I(1)\) process) is one where \(\Delta X_t = X_t - X_{t-1}\) is covariance stationary

    • Integrated process of order 2 (\(I(2)\) process) is one where \(\Delta^2 X_t = \Delta X_t - \Delta X_{t-1}\) is covariance stationary (but \(\Delta X_t\) is not)

Dickey-Fuller test can help distinguish between trend and difference stationary processes

  • Regresses the first difference of the observations onto the preceding observation and a time trend:

    e.g. \(\Delta X_t = \alpha_0 + \alpha_1t + \alpha_2X_{t-1} + \epsilon_t\)

    • Test statistic \(\alpha_2 \big/ s.e.(\alpha_2)\) is compared to the critical values determined by Dickey-Fuller

    • If \(\alpha_2\) is (statistically) significantly different to 0 then the series is an integrated process (else trend stationary)

ARCH and GARCH

Models for series that exhibit heteroskedasticity

ARCH

Autoregressive conditional heteroskedasticity (ARCH)

  • Based on strictly stationary white noise process (\(Z_t\)) with 0 mean and unit s.d.

  • ARCH process is constructed so that the s.d. (\(\sigma_t\)) varies over time

Definition of \(ARCH(p)\) process

The process \(\{X_t : t= ..., -2, -1, 0,1,2,...\}\) is an \(ARCH(p)\) process if:

\(X_t = \epsilon_t = \sigma_t Z_t\)

  • \(\sigma_t^2 = \alpha_0 + \sum \limits_{i=1}^p \alpha_i X_{t-i}^2\)

  • \(\alpha_0 > 0\)

  • \(\alpha_i \geq 0\) for \(i = 1,...,p\)

  • \(\{Z_t : t = ...,-2, -1, 0, 1, 2, ...\}\) is strict white noise with mean 0 and variance 1

Heteroskedasticity and volatility clustering

  • Given the \(\{X_t\}\) is weakly stationary

    \(\hookrightarrow\) \(\mathrm{E}(X^2_t) < \infty\) and

    \(\hookrightarrow\) \(\mathrm{Var}\left(X_t \mid F_{t-1}\right) = \mathrm{E}\left(\sigma_t^2 Z_t^2 \mid F_{t-1}\right) = \sigma^2_t \mathrm{Var}(Z_t) = \sigma^2_t = \alpha_0 + \sum \limits_{i=1}^p \alpha_i X_{t-i}^2\)

    • \(F_{t-1}\) is the history of the process up to time \(t-1\)

  • Conditional variance of \(\{X_t\}\) is changing over time

    • Property of heteroscedasticity

    • Function of the previous squared values of the process

    • Large change in the value of the process is often followed by a period of high volatility

      \(\therefore\) Model incorporates the feature of volatility clustering

  • Although the conditional variance is changing, we don’t know in advance when periods of high and low future volatility will occur

    \(\therefore\) Still possible for the \(ARCH\) process to be stationary

Condition for weakly stationary

  • \(ARCH(1)\) is weakly (covariance) stationary white noise process iff \(\alpha_1 < 1\)

    Variance of the process is \(\dfrac{\alpha_0}{1-\alpha_1}\)

  • Condition for \(ARCH(p)\) to be weakly stationary

    Roots of the polynomial \(f(z) = 1 - \alpha_1 z - \alpha_2 z^2 - \dots - \alpha_p z^p = 0\) must lie outside the unit circle

Condition for strictly stationary

  • Result for weak stationarity of an \(ARCH(1)\) process above does not depend on the distribution of \(Z_t\) but the result for strict stationarity does

  • If \(Z_t \sim N(0,1)\) then the condition for a strictly stationary solution is approximately \(\alpha_1 < 2e^{\eta} \approx 3.562\) where \(\eta\) is the Euler-Mascheroni constant (= 0.57721…, usually denoted by gamma)

  • For \(m\geq 1\) the strictly stationary \(ARCH(1)\) process has finite moments of order \(2m\) iff \(\mathrm{E}(Z^{2m}_t) < \infty\) and \(\alpha_1 < \left[\mathrm{E}(\left(Z_t^{2m} \right)\right]^{-\frac{1}{m}}\)

    And the XS kurtosis is \(\kappa = \dfrac{\mathrm{E}(Z^4_t)(1-\alpha^2_1)}{1-\alpha^2_1\mathrm{E}(Z^4_t)} - 3\)

GARCH

Generalized ARCH

  • Volatility is now allowed to depend on previous values of volatility as well as previous values of the process

  • Periods of high volatility tend to last for a long time

    • Also the case for high order \(ARCH\) but \(GARCH\) produce the effect with a lower number of parameters

Definition

The process \(\{X_t : t = ..., -2, -1, 0, 1, 2, ...\}\) is a \(GRACH(p,q)\) process if:

\(X_t = \sigma_t Z_t\)

  • \(\sigma_i = \sqrt{\alpha_0 + \sum \limits_{i=1}^p \alpha_i X^2_{t-i} + \sum \limits_{j=1}^q \beta_j \sigma^2_{t-j}}\)

  • \(\alpha_0 > 0\)

  • \(\alpha_i \geq 0\) for \(i = 1,...,p\)

  • \(\beta_j \geq 0\) for \(j = 1,...,q\)

  • \(\{Z_t : t = ...,-2, -1, 0, 1, 2, ...\}\) is strict white noise with mean 0 and variance 1

Condition for weakly stationary

  • \(GARCH(1,1)\) is weakly stationary iff \(\alpha_1 + \beta_1 <1\)

    Variance of the process is \(\dfrac{\alpha_0}{1-\alpha_1 - \beta_1}\)

  • Condition for \(GARCH(p,q)\) to be weakly stationary

    If \(\sum \limits_{i=1}^p \alpha_i + \sum \limits_{j=1}^q \beta_j <1\)

Integrated GARCH

IGARCH

  • Occurs when :

    \(\sum \limits_{i=1}^p \alpha_i + \sum \limits_{j=1}^q \beta_j = 1\)

  • For an \(IGRACH(1,1)\) process \(\beta_1 = 1 - \alpha_1\)

    \(\hookrightarrow\) \(\Delta X_t^2 = \alpha_0 - (1- \alpha_1)V_{t-1} + V_t\)

  • Similar to \(ARIMA(0,1,1)\) model for \(\{X_t^2\}\)

    Although \(\{V_t\}\) is not white noise

ARMA model with GARCH errors

  1. First fit the model to \(ARMA\)

  2. Then fit a \(GARCH\) model to the resulting residuals (if residuals are not white noise)

Definition

\(\{X_t\}\) is an \(ARMA(p,q)\) with \(GARCH(r,s)\) errors if:

  • Weakly stationary

  • \(X_t = \mu_t + \sigma_t Z_t\) where \(\{Z_t\}\) is strict white noise with mean 0 and variance 1

  • \(\mu_t = \mu + \sum \limits_{i=1}^p \varphi_i(X_{t-i} - \mu) + \sum \limits_{j=1}^q \theta_j (X_{t-j} - \mu_{t-j})\)

  • \(\sigma_t^2 = \alpha_0 + \sum \limits_{i=1}^r \alpha_i (X_{t-i} - \mu_{t-i})^2 + \sum \limits_{j=1}^s \beta_j \sigma^2_{t-j}\)

With:

  • \(\alpha_0 >0\), \(\alpha_i \geq 0\) for \(i = 1,2,...,r\)

  • \(\beta_j \geq 0\) for \(j=1,2,...,s\)

  • \(\sum \limits_{i=1}^r \alpha_i + \sum \limits_{j=1}^s \beta_j < 1\)

Fitting GARCH

Most commonly used technique is the method of maximum likelihood

Challenge

  1. PDF of \(X_0\) is not known

    We get around this by considering the conditional likelihood given \(X_0 = x_0\)

  2. \(\sigma_0\) is not actually observed

    We can choose a starting value for it, which might be the sample s.d. of \(X_1,...,X_T\) or just 0

Testing the fit

As with \(ARMA\) it is good to check the goodness of fit of a GARCH model by examining the residual

  • e.g. for \(ARAM\) with \(GARCH\) errors, the \(GARCH\) errors will be of the form \(X_t = \epsilon_t = \sigma_t Z_t\)

Must distinguish between unstandardized and standardized residuals

  • Unstandardized residuals:

    Residuals \(\hat{\epsilon}_1,...,\hat{epsilon}_T\) after having fitted the \(ARMA\) part of the model

    • If the \(ARMA\) model and parameteriztion are correct then the unstandardized residuals should look like a pure \(GARCH\) process

  • Standardized residuals:

    Reconstructed realizations of the strict white noise process \(\{Z_t\}\) that is assumed to drive the \(GARCH\) part of the model

    Calculated using:

    \(\hat{Z}_t = \dfrac{\hat{\epsilon_t}}{\hat{\sigma}_t}\)

    Where: \(\hat{\sigma^2_t} = \hat{\alpha}_0 + \sum \limits_{i=1}^{p_2} \hat{\alpha}_i \hat{\epsilon}_{t-i}^2 + \sum \limits_{j=1}^{q_2} \hat{\beta}_j\hat{\sigma}_{t-j}^2\)

    Need initial values to use the equations above

    • Can set starting \(\hat{\epsilon}_t =0\) and \(\hat{\sigma}_t = 0\) (or sample s.d. of the observed time series)

    The standardized residuals should behave like strict white noise

    • Test with correlograms and statistical test (portmanteau test and the turning point test)

    • If there is insufficient evidence to reject the hypothesis of strict white noise, the validity of the distribution used to construct the likelihood function (i.e. \(f\), the PDF of \(Z_t\)) can be investigated using QQ plots and goodness of fit test for the normal or scaled t distribution

GARCH Based Volatility Prediction

Data: \(X_{t-n+1},...,X_t\) from a \(GARCH\) model

Predict: Values of \(\sigma_{t+h}\) for \(h \geq 1\) from a fitted \(GARCH(1,1)\) model

Step 1

  • Let \(F_t\) be the infinite history of the process up to time \(t\)

  • Assume that the model is weakly stationary

    \(\hookrightarrow\) \(\mathrm{E}(X^2_t) = \mathrm{E}(\sigma^2_t) < \infty\):

Then:

\(\begin{align} \mathrm{E}(X^2_{t+1} \mid F_t) &= \mathrm{E}(\sigma^2_{t+1}Z^2_{t+1} \mid F_t) \\ &= \mathrm{E}\left[(\alpha_0 + \alpha_1 x^2_t +\beta_1 \sigma_t^2)Z^2_{t+1} \mid F_t\right] \\ &= (\alpha_0 + \alpha_1 x^2_t + \beta_1\sigma_t^2)\mathrm{E}(Z^2_{t+1} \mid F_t) \\ &= \alpha_0 + \alpha_1 x^2_t + \beta_1 \sigma^2_t \\ \end{align}\)

  • Since \(\mathrm{E}(Z^2_{t+1} \mid F_t) = \mathrm{E}(Z^2_{t+1}) = \mathrm{Var}(Z^2_{t+1}) + [\mathrm{E}(Z_{t+1})]^2 = 1\)

  • Also note \(\mathrm{E}(X^2_{t+1} \mid F_t) = \sigma^2_{t+1}\)

Step 2

  • Problem with the above is that we don’t have infinite history of the process

    • For \(\sigma^2_t\):

      Approximate using the residual equations in the GARCH section

    • For \(\sigma^2_{t+1}\):

      Estimate with the approximate forecast of \(X_{t+1}^2\)

      \(\hat{\sigma}^2_{t+1} = \hat{\mathrm{E}}(X_{t+1}^2 \mid F_t) = \alpha_0 + \alpha_1 X^2_t + \beta_1 \hat{\sigma}^2_t\)

    • Use above equation to estimate volatility one step ahead recursively

  • To estimate \(X_{t+h}\) and \(\sigma_{t+h}\) given the information up to \(t\)

    Both \(X_{t+h}\) and \(\sigma_{t+h}\) are r.v. and:

    \(\begin{align} \mathrm{E}(X^2_{t+h} \mid F) &= \mathrm{E}(\sigma^2_{t+h} \mid F_t) \\ &= \alpha_0 + \alpha_1 \mathrm{E}(X^2_{t+h-1} \mid F_t) + \beta_1 \mathrm{E}(\sigma^2_{t+h-1} \mid F_t) \\ &= \alpha_0 + (\alpha_1 + \beta_1)\mathrm{E}(X^2_{t+h-1} \mid F_t) \\ \end{align}\)

  • So in general:

    \(\mathrm{E}(X^2_{t+h} \mid F) = \alpha_0 \sum \limits_{i=0}^{h-1}(\alpha_1 + \beta_1)^i + (\alpha_1 + \beta_1)^{h-1} (\alpha_1 X^2_t + \beta_1 \sigma^2_t)\)

    • Use this formula by replacing \(\sigma^2_t\) by \(\hat{\sigma}^2_t\)

Data Frequency

Problem with time series: insufficient data may be available to generate statistics based on a long timescale (e.g. annual volatility)

  • In such case we may need to scale up statistics

    (e.g. estimate annual volatility from monthly volatility)

  • If data is available from \(N\) groups:

    Each of which is iid normal with:

    • Means \(\bar{\mathbf{X}}\)

    • Covariances given by the \(N \times N\) matrix \(\boldsymbol{\Sigma}\)

    \(\hookrightarrow\) means and covariances working over a timescale \(T\) times as long are:

    • \(T \bar{\mathbf{X}}\) and

    • \(T\boldsymbol{\sigma}\)

  • E.g. if data is available based on a monthly timescale then multiplying the means and covariance by 12 will provide the corresponding annual statistics

  • Similarly if \(\bar{\mathbf{X}} = \mathbf{0}\)

    \(\hookrightarrow\) Aggregate volatility for the longer timescale can be derived as:

    \(\sqrt{T}\) multiplied by the corresponding volatility for the shorter timescale

  • Applying to VaR

    Annual VaR might be derived from the monthly VaR by multiplying by \(\sqrt{12}\)
    (i.e. the square root of time rule)

Such scaling becomes inaccurate if the data is not iid normal

  • e.g. if its skewed or there is serial correlation

  • Better approach in such situations is to parameterize as stochastic model

    Using the shorter timescale data and then to derive the longer timescale statistics from a set of simulations