class: center, middle, inverse, title-slide # Nonlinear times series modelling ### Kevin Kotzé --- <!-- layout: true --> <!-- background-image: url(image/logo.svg) --> <!-- background-position: 2% 98% --> <!-- background-size: 10% --> --- # Contents 1. Introduction 1. Threshold Autoregressive (TAR) models 1. Smooth Transition Autoregressive (STAR) models 1. STAR: Parameter estimation 1. Testing - nonlinearity 1. STAR: Testing - diagnostics 1. STAR example 1. Markov Switching Models 1. Artificial Neural Network models 1. Conclusion --- # Introduction - Econometrics: use available information to describe relationships - Time series usually extend over quite a long period of time - Over such a period we often incur certain changes that influence the behaviour of the DGP (e.g. expansion-recession, etc.) - Regime switching models incorporate the dynamic state dependent behaviour of economic or financial variables --- background-image: url(image/graph1.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure : Regime switching DGPs --- # Further examples - Further examples of such behaviour occur in stock markets - Skewness (large negative returns are more common than large positive returns) and - Kurtosis (large absolute returns are frequently observed) - Other uses: - determination of the actual change - multiple equilibria - Other nonlinear models (GAR, bilinear,etc) --- # Dummy variable models - Assume the sample can be split into separate groups (regimes) - Parameters are constant within the groups (regimes) but differ between groups - There is no transition period - Identification of the groups (regimes) is known with certainty in advance - Regime switching process is deterministic --- # Dummy variable models - Dummy removes that part of the residual that was generated as a result of a sudden change - Facilitates more accurate parameter estimation - Chow break test used to determine if break is significant - Split sample into groups and test null hypothesis of constant coefficients in subgroups - Could also use CUSUM test - Tests for possible parameter instability in sample --- # Basic Regime Switching - Regime is described by a stochastic process - Future regimes are not know with certainty - regime is determined by an observable variable (past & present known) - regime is determined by an unobserved stochastic process (assign probabilities to regime) - Simple specification `\begin{eqnarray} y_t = \left\{ \begin{array}{cc} \phi_{0,1} + \phi_{1,1} x_t + \varepsilon_{t} \;\;\;\;\;\;\;\;\; \text{in regime one} \\ \phi_{0,2} + \phi_{1,2} x_t + \varepsilon_{t} \;\;\;\;\;\;\;\;\; \text{in regime two} \end{array} \right. \end{eqnarray}` - where `\(\varepsilon_t \sim(0,\sigma^2_{i})\)` in regime `\(i, i = 1, 2\)` --- # Basic Regime Switching - In contrast with dummy variable models: - Each regime is modeled explicitly - May have different errors for each regime % may be of relevance when interpreting the individual shocks from each regime - Important when interpreting the model parameters - Relevant for identifying the current regime --- # Threshold Autoregressive (TAR) model - Regime can be described by observable variable, `\(q_t\)` , relative to a threshold, `\(c\)` `\begin{eqnarray} y_t = \left\{ \begin{array}{cc} \phi_{0,1} + \phi_{1,1} x_t + \varepsilon_{t} \;\;\;\;\;\;\;\;\; \text{if } q_t \leq c \\ \phi_{0,2} + \phi_{1,2} x_t + \varepsilon_{t} \;\;\;\;\;\;\;\;\; \text{if } q_t > c \end{array} \right. \end{eqnarray}` - where `\(\varepsilon_t \sim(0,\sigma^2_{i})\)` in regime `\(i, i = 1, 2\)` --- # Threshold Autoregressive (TAR) model - Consider Shen & Hakes (1995) - reaction function of the Taiwanese central bank - inflation determines which regime they are in - targeted inflation rate provides the threshold - `\(F\)`-test for nonlinearity - no inflation: pursue output growth and low inflation - low inflation: pursue output growth (with no response to inflation) - moderate & high inflation: pursue only inflation and not output growth --- # SETAR models - Self Extracting Threshold Autoregressive models - Observable variable, `\(q_t\)` , is a lagged value of the series itself - Hence, for `\(AR(1)\)` SETAR model; `\begin{eqnarray} y_t = \left\{ \begin{array}{cc} \phi_{1,1} y_{t-1} + \varepsilon_{t} \;\;\;\;\;\;\;\;\; \text{if } y_{t-1} \leq c \\ \phi_{1,2} y_{t-1} + \varepsilon_{t} \;\;\;\;\;\;\;\;\; \text{if } y_{t-1} > c \end{array} \right. \end{eqnarray}` - where `\(\varepsilon_t \sim(0,\sigma^2_{i})\)` in regime `\(i, i = 1, 2\)` - such that `\(\varepsilon_{1t}\)` and `\(\varepsilon_{2t}\)` are responsible for the regime switching --- # SETAR models - Alternative representation - If `\(\sigma^2_1 = \sigma^2_2 = \sigma^2\)`, then the model may be written as; `\begin{eqnarray} y_t = (\phi_{0,1} + \phi_{1,1} y_{t-1})(1-I\left[y_{t-1} > c \right] ) \ldots \\ \ldots + (\phi_{0,2} + \phi_{1,2} y_{t-1})(I\left[y_{t-1} > c \right] ) + \varepsilon_t \end{eqnarray}` - where `\(I[\cdot]\)` is an indicator function with `\(I[\cdot]= 1\)` if event `\(1\)` occurs and `\(I[\cdot] = 0\)` otherwise --- # STAR models - Smooth Transition Autoregressive models - More gradual transition between regimes - Weight series with a continuous (logistic) function - Where `\(\gamma\)` is the smoothing parameter, `\(\gamma > 0\)` `\begin{eqnarray} G\left[q_t ; \gamma, c \right] \end{eqnarray}` - Changes smoothly from `\(0\)` to `\(1\)` as `\(q_t\)` increases - A popular choice for the transition mechanism is the logistic function, such that; `\begin{eqnarray} G\left[q_t ; \gamma, c \right] = \frac{1}{1+\exp (-\gamma [q_t - c])} \end{eqnarray}` --- background-image: url(image/logf1.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure : Logistic function for different `\(\gamma\)` - As `\(\gamma \rightarrow \infty\)`, STAR model represents a TAR model - As `\(\gamma \rightarrow 0\)`, STAR model represents a linear model --- # STAR models - The resulting expression for the model is; `\begin{eqnarray} y_t = \phi_{1} x_{t}(1-G\left[q_t ; \gamma, c \right] ) + \phi_{2} x_{t}(G\left[q_t ; \gamma, c \right] ) + \varepsilon_t \end{eqnarray}` - where `\(\varepsilon_t \sim(0,\sigma^2)\)` and `\(\gamma\)` is the smoothing parameter in the continuous function --- # STAR: Parameter estimation - Nonlinear least squares is often used to estimate parameters - Optimization technique that seeks to minimize an objective function with the aid of an iterative search process - Consider the standard linear model the estimated parameters and residual sum of squares are calculated as: `\begin{eqnarray} y_t &= &E(y_t | \mathbf{x}_t ) + \varepsilon_t \\ &= & \phi_1 x_{t1} + \phi_2 x_{t2} + \ldots + \phi_p x_{tp} + \varepsilon_t \end{eqnarray}` --- # STAR: Parameter estimation - The estimated parameters and residual sum of squares are calculated as: `\begin{eqnarray} \hat{\phi} & = &(\mathbf{x}^{\prime}\mathbf{x})^{-1} \mathbf{x}^{\prime}\mathbf{y}\\ S(\hat{\phi} )& = &\sum_{t=1}^{n} \left[ y_t - \hat{\phi}_{1} x_{t1} - \hat{\phi}_{2} x_{t2} - \ldots - \hat{\phi}_{p} x_{tp} \right]^2 \end{eqnarray}` - Least squares estimation involves identifying the parameter values that minimize the sum of square residuals --- # STAR: Parameter estimation - Could also have found these parameters with an iterative search technique: `\begin{eqnarray} S(\hat{\beta} ) = \sum_{t=1}^{n} \left[ y_t - \tilde{\phi}_{1} x_{t1} - \tilde{\phi}_{2} x_{t2} - \ldots - \tilde{\phi}_{p} x_{tp} \ldots\right. \\ \left. \ldots - (\hat{\phi}_1 - \tilde{\phi}_1)x_{t1} - (\hat{\phi}_2 - \tilde{\phi}_2)x_{t2} - \ldots - (\hat{\phi}_p - \tilde{\phi}_p)x_{tp}\right]^2 \end{eqnarray}` - where the initial guess for the value of the coefficients in matrix `\(\phi\)` is expressed as `\(\tilde{\phi}\)` --- # STAR: Parameter estimation - Why is this means of parameter estimation so useful? - Parameters need not only include estimates for `\(\phi\)` - In the STAR model the parameters could include `\(\mathbf{\psi} = (\phi ; G\left[q_t ; \gamma ,c\right])\)` - Such that `\begin{eqnarray} y_t = f(\mathbf{x_t}, \mathbf{\psi}) + \varepsilon_t \;\;\;\;\;\;\;\;\; t=1,2,\ldots,n \end{eqnarray}` - where `\(\mathbf{\psi} = (\psi_1, \psi_2, \ldots , \psi_p)\)` represents the parameters in the nonlinear regression, including `\(\mathbf{\psi} = (\phi ; G\left[q_t ; \gamma ,c\right])\)` --- # STAR: Testing - nonlinearity - Before estimating a regime switching model test for nonlinearity - Unfortunately there is no simple test which indicates the form of nonlinearity - Likelihood Ratio (LR) test: based on the loss of log-likelihood following the imposition of certain restrictions (i.e. linearity) - Requires estimates for both linear and nonlinear models - Large test statistic relative to critical value from `\(\chi^2\)` distribution = reject null - Some researchers make use of LM type tests - No test is going to tell you which form of nonlinearity is the correct *a priori* --- # STAR: Testing - diagnostics - Not all test statistics of linear models are applicable - Test for serial correlation in each regime `\begin{eqnarray} y_t = (\phi_{0,1} + \phi_{1,1} y_{t-1})(1-I\left[y_{t-1} > c \right] ) \ldots \\ \ldots + (\phi_{0,2} + \phi_{1,2} y_{t-1})(I\left[y_{t-1} > c \right] ) + \varepsilon_t \end{eqnarray}` - Could use some form of LR test but this would be time intensive - Would also need to test for how many regimes to include --- # STAR: Testing - diagnostics - Testing for remaining nonlinearity: - LM statistic testing the null a 2 regime STAR model capture the nonlinearity (i.e. against the alternative of whether a 3 regime STAR) - Testing for parameter constancy: - investigate whether we would need to include time varying parameters - test hypothesis `\(\gamma_2 = 0\)` against the alternative of smoothly changing parameters - Very few really good in-sample tests - hence most people use extensive out-of-sample tests --- # STAR example - Apply a STAR model to describe USDZAR exchange rate: - reject the null of linearity using LM test - but it does not tell us what form of nonlinearity to use - estimate a two regime STAR model for an `\(AR(3)\)` process - threshold variable is the average exchange rate for the last 4 weeks - uses NLS optimization procedure to find `\(\phi\)` and `\(\gamma\)` parameters - starting values are determined with simplified grid search technique --- # STAR: Example `\(\;\)` | Coef | Std.Err | t-val | prob | -----------|----------|----------|---------|---------| `\(\phi\)` 1.1 | 0.42437 | 0.20515 | 2.0686 | 0.03859 | `\(\phi\)` 1.2 | 0.47478 | 0.60845 | 0.7803 | 0.4352 | `\(\phi\)` 2.1 | -0.22728 | 3.63697 | -0.0625 | 0.95017 | `\(\phi\)` 2.2 | -2.31202 | 26.49029 | -0.0873 | 0.93045 | `\(\phi\)` 2.3 | 1.0281 | 9.9477 | 0.1034 | 0.91769 | `\(\gamma\)` | 8.61485 | 5.52041 | 1.5605 | 0.11863 | thresh | 0.15996 | 1.04446 | 0.1532 | 0.87828 | AIC = -748\\ p = 0.0013 (AR versus STAR) --- # Specification for regime switching models - Granger (1993): Building models of nonlinear relationships are inherently more difficult than linear ones. There are *more possibilities*, many *more parameters* and thus *more mistakes* can be made. It is suggested that a strategy be applied when attempting such modelling involving *testing for linearity*, considering just a *few model types* of *parsimonious* form and performing *post-sample evaluation* of the models compared to a linear one. The strategy proposed is a 'simple-to-general' one and the application of a heteroscedasticity correction is not recommended. --- # Specification procedures for regime switching models - Specify a linear model to describe `\(y_t\)` in terms of `\(x_t\)` - Test the null hypothesis of linearity against the alternative of TAR, STAR, MSW, or ANN nonlinearity - portmanteau, RESET and Macleod-Li tests - testing the null of linearity in LM type tests - existence of unidentified nuisance parameters `\((c\)` and `\(\gamma)\)` - simulation - Estimate the parameters in the selected model - Evaluate the model using diagnostic tests - LM type tests: serial correlation, parameter consistency, heteroscedasticity, omitted variables - Modify the model if necessary - Use the model for descriptive or forecasting purposes --- # Markov Switching Models - Not able to observe reliable variable that we could use as the regime indicator, `\(q_t\)` - Regime at `\(t\)` derived from unobserved process, `\(S_t\)` - `\(S_t\)` represents the probability of being in a certain state at a point of time `\begin{eqnarray} y_t = \left\{ \begin{array}{cc} \phi_{0,1} + \phi_{1,1} x_t + \varepsilon_{t} \;\;\;\;\;\;\;\;\; \text{if } S_t =0 \\ \phi_{0,2} + \phi_{1,2} x_t + \varepsilon_{t} \;\;\;\;\;\;\;\;\; \text{if } S_t =1 \end{array} \right. \end{eqnarray}` - where `\(\varepsilon_t \sim \mathcal{N} (0,\sigma^2_{S_t})\)` in regime `\(i, i = 1, 2\)` --- background-image: url(image/markov.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure : Markov Chain --- # Markov Switching Models - Unobserved process is first order Markov process - Current regime only depends on previous regime and `\(p_{ij}\)` - System will either be in regime `\(0\)` or regime `\(1\)` - At `\(t\)` there is a probability `\(p_{ij}\)` that the system, if in regime `\(i\)` will change to regime `\(j\)` (where `\(i, j =1,0\)`) --- # Fixed transition probabilities - Hence; `\begin{eqnarray} P (s_t = 0 | s_{t - 1} = 0) & = & p_{00}\\ P (s_t = 1 | s_{t - 1} = 0) & = & p_{01} = 1 - p_{00}\\ P (s_t = 0 | s_{t - 1} = 1) & = & p_{10} = 1 - p_{11}\\ P (s_t = 1 | s_{t - 1} = 1) & = & p_{11} \end{eqnarray}` - where `\(p_{10} + p_{11} = 1\)` and `\(p_{00} + p_{01} = 1\)` - If `\(p_{10} = p_{01} = 1\)` and `\(p_{00} = p_{11} = 0\)`, then the system will be in continuously change - If `\(p_{10} = p_{01} = 0\)` and `\(p_{00} = p_{11} = 1\)`, then the system will never change out of one regime - Conditional states! --- # Steady states - If all the probabilities are nonzero then the system will approach a stable point where; `\begin{eqnarray} P[S_0] = \frac{1-p_{11}}{2-p_{00}-p_{11}} & \;\;\; \text{or} \;\;\; & \frac{p_{10}}{p_{01}+ p_{10}}\\ P[S_1] = \frac{1-p_{00}}{2-p_{00}-p_{11}}& \;\;\; \text{or} \;\;\; & \frac{p_{01}}{p_{01}+ p_{10}} \end{eqnarray}` - These steady state probabilities describe the unconditional probabilities of being in each regime at a point in time - Although the expected duration of being in a particular regime can differ, the probabilities are forced to be constant over time - Hence - fixed transition probability MSW --- # Time varying transition probabilities - May want transitional probabilities to vary over time to include a richer information set. For example, - economy in robust recovery is less likely to fall into a recession - Make `\(p_{ij}\)` a function of duration or a function of another variable - Hence; `\begin{eqnarray} P (s_t = 0 | s_{t - 1} = 0, \psi_t) & = & p_{00}(\psi_t)\\ P (s_t = 1 | s_{t - 1} = 0, \psi_t) & = & p_{01}(\psi_t) \\ P (s_t = 0 | s_{t - 1} = 1, \psi_t) & = & p_{10}(\psi_t)\\ P (s_t = 1 | s_{t - 1} = 1, \psi_t) & = & p_{11}(\psi_t) \end{eqnarray}` - where `\(\Omega_t\)` is the information set for evolution of the unobserved regime --- # Varying TVTP - Could allow for transitional probabilities to be dependent on the value of an exogenous variable, `\(z_t\)`, in a logistic function: `\begin{eqnarray} p_{00} = P(S-t =0 | S_{t-1} =0) = \frac{\exp (\alpha_0 + \beta_0 z_t)}{(1+ \exp (\alpha_0 + \beta_0 z_t)}\\ p_{11} = P(S-t =1 | S_{t-1} =1) = \frac{\exp (\alpha_1 + \beta_1 z_t)}{(1+ \exp (\alpha_1 + \beta_1 z_t)} \end{eqnarray}` --- # Parameter estimation - Procedure for the Markov Switching model is non-standard since it seeks to obtain; - estimates of the parameters in the different regimes - estimates for the probability of transition over time - estimates for the probability of being in a particular state at a period of time - Draw observed variable `\(y_t\)` from distribution conditional on the discrete random variable `\(S_t\)` to obtain; `\begin{eqnarray} f(y_t | S_t , \Omega_{t-1}) \end{eqnarray}` - Assume the unobserved process is generated by a probability distribution `\begin{eqnarray} f(y_t | S_t , \Omega_{t-1}) = P[S_t =j|\Omega_{t-1}] \end{eqnarray}` --- # Parameter estimation - Conditional probability from joint probability; `\begin{eqnarray} P(A\; \text{and } B) = P(A|B) \cdot P(B) \end{eqnarray}` - To determine the probability that both `\(S_t = j\)` and `\(y_t\)` falls within a predetermined interval, conditional probability is given; `\begin{eqnarray} f(y_t , S_t | \Omega_{t-1}) &=& f(y_t | S_t , \Omega_{t-1})\cdot f(S_t | \Omega_{t-1})\\ &=& f(y_t | S_t , \Omega_{t-1}) \cdot P[S_t =j|\Omega_{t-1}] \end{eqnarray}` - where `\(\Omega_{t -1}\)` refers to the information set to `\(t - 1\)` and $j = 0, 1, \ldots $ --- # Hamilton Filter (1989) - Algorithm responsible for calculating the probability that the process is in regime `\(j\)` at time `\(t\)` given; - all observations up to time `\(t - 1\)` (i.e. the forecast) - all observations up to time `\(t\)` (i.e. the inference) - all observations in the entire sample (i.e. the smoothed inference) - Works similar to the Kalman filter, which may be used to produce values for the unobserved process --- # Basic procedure - The following iterative procedure is suggested: - obtain starting values for the model parameters - compute the smoothed regime probabilities with the aid of the procedures specified in the forecast & inference sections - combine these estimates with the initial estimates of the transition probabilities to obtain new estimates for the transition probabilities working backwards from `\(n\)` to `\(1\)` - calculate values for the remaining `\(\phi\)` parameters - iterating this procedure renders a new set of estimates until convergence occurs --- # Forecasting - There are no shocks in the out-of-sample period - Could assume that economy/state persists over the out-of-sample period - Could generate forecasts under each regime and combine them with the transition probabilities --- # Applications of Markov Switching - Business cycle analysis - determination of turning points - determination of length of business cycle - forecasting turning points - Appreciation and depreciation regimes in exchange rates - Different regimes in volatility - Different regimes in interest rates - Different political regimes and other institutional characteristics --- # Hamilton's Markov Switching model - Growth in U.S. Real GNP follows an `\(AR(4)\)` process - Two state Markov switching (expansion & recession) - Hence; `\begin{eqnarray} (y_t - \mu_{S_t}) = \phi_1 (y_{t-1} - \mu_{S_{t-1}}) + \phi_2 (y_{t-2} - \mu_{S_{t-2}}) \ldots\\ \ldots + \phi_3 (y_{t-3} - \mu_{S_{t-3}}) + \phi_4 (y_{t-4} - \mu_{S_{t-4}}) + \varepsilon_t \end{eqnarray}` - where `\(\varepsilon_t = \mathsf{i.i.d.} \mathcal{N}(0,\sigma^{2}_{S_t})\)` --- background-image: url(image/MSW1.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure: Hamilton's Markov Switching model - Where the shaded areas represent the NBER business cycles, this model was able to replicate the business cycles relatively well for the given sample period --- # Moolman's Markov Switching model - Moolman (2004) derives a model with TVTP, which are influenced by yield spreads - Once again growth in S.A. Real GDP follows an `\(AR(4)\)` process - Two state Markov switching (expansion & recession) - Hence; `\begin{eqnarray} y_t = \mu_2 (1-S_t) + \mu_1S_t + \phi_1 (y_{t-1} -(\mu_2 (1-S_{t-1})) + \mu_{S_{t-1}}) + \ldots\\ \ldots + \phi_4 (y_{t-4} - (\mu_2 (1-S_{t-1})) + \mu_{S_{t-4}}) + \varepsilon_t \end{eqnarray}` --- background-image: url(image/MOOL1.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure : Moolman's Markov Switching model --- # Engle's Markov Switching model - Engle and Hamilton (1990) showed that the US Dollar exchange rate appears to follow long swings as it drifts upwards (downwards) for a considerable period of time - Engle (1994) showed that for the exchange rates of 18 different countries; - MSW model outperforms random walk models on in-sample testing - Does not outperform the random walk model or the forward exchange rate in its out-of-sample forecasting ability - Also suggested that the model picks up a change in regime fairly early on, however, this feature seems to be dependent upon the persistence of a regime --- # Artificial Neural Network models - Often regarded as flexible non-parametric models that can approximate any nonlinear function arbitrarily closely - Could also be specified as flexible regime switching models and could be interpreted as such - Drawback: - parameters are impossible to interpret, hence, only used for pattern recognition and forecasting - superior in sample fit could be the result of modelling irregular / unpredictable noise --- background-image: url(image/ANNdgp.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure : Scatter plot for hypothetical relationship, `\(x_t\)` and `\(y_t\)` --- background-image: url(image/ann1.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure : ANN model for time series with `\(q=3\)` --- # Estimation - The idea is similar to a STAR model - For example, single hidden-layer feedforward ANN takes the form; `\begin{eqnarray} y_t = \phi_0 + \sum_{j=1}^{1} \beta_j G\left( \gamma_j \left(x_t - c_j \right)\right) + \varepsilon_t \end{eqnarray}` - Where the logistic function is used for smoothing `\(G(\cdot)\)`; `\begin{eqnarray} G(z) = \frac{1}{1+\exp(-z)} \end{eqnarray}` --- # Estimation - Assume parameters `\((c_j , j = 1, \ldots , q)\)` follow, `\(c_1 \leq c_2 \leq \ldots \leq c_q\)` - Then; `\begin{eqnarray} \hat{g}(x_t) = \left\{ \begin{array}{ll} \phi_{0} & \text{if } x_t \leq c_1 \\ \phi_{0} + \beta_1 & \text{if }c_1 \leq x_t \leq c_2 \\ \phi_{0} + \beta_1 + \beta_2 & \text{if }c_2 \leq x_t \leq c_3 \\ \vdots & \\ \phi_{0} + \beta_1 + \beta_2 + \ldots + \beta_q & \text{if }c_q < x_t \\ \end{array} \right. \end{eqnarray}` --- # Difference to STAR - Although looks similar to a STAR model it is different; - STAR: regime is usually determined by `\(1\)` lagged value of `\(y_t\)`, - ANN: regime normally considers `\(p\)` lagged values of `\(y_t\)` - STAR: each regime has its own intercept - ANN: only one intercept is used - An important difference is ANN models normally use more than one logistic function, which gives it the ability to approximate any nonlinear model arbitrarily closely --- background-image: url(image/ann_nom.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure : Traditional (alternative) nomenclature --- # Testing - Evaluate the model testing in-sample fit - Conduct various misspecification tests for remaining nonlinearity and parameter constancy - Conduct residual diagnostic tests for serial correlation, heteroscedasticity and normality - Conduct stringent out-of-sample testing - Granger's (1993)