## Glossary

This glossary gives brief definitions of all the key terms used in the book.

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | Y

## A

**adjusted **** R**2: a measure of how well a model fits the sample data that automatically penalises models with large numbers of parameters.

**Akaike information criterion (AIC)**: a metric that can be used to select the best fitting from a set of competing models and that incorporates a weak penalty term for including additional parameters.

**alternative hypothesis**: a formal expression as part of a hypothesis testing framework that encompasses all of the remaining outcomes of interest aside from that incorporated into the null hypothesis.

**arbitrage**: a concept from finance that refers to the situation where profits can be made without taking any risk (and without using any wealth).

**asymptotic**: a property that applies as the sample size tends to infinity.

**autocorrelation**: a standardised measure, which must lie between −1 and +1, of the extent to which the current value of a series is related to its own previous values.

**autocorrelation function**: a set of estimated values showing the strength of association between a variable and its previous values as the lag length increases.

**autocovariance**: an unstandardised measure of the extent to which the current value of a series is related to its own previous values.

**autoregressive conditional heteroscedasticity (ARCH) model**: a time series model for volatilities.

**autoregressive (AR) model**: a time series model where the current value of a series is fitted with its previous values.

**autoregressive moving average (ARMA) model**: a time series model where the current value of a series is fitted with its previous values (the autoregressive part) and the current and previous values of an error term (the moving average part).

**autoregressive volatility (ARV) model**: a time series model where the current volatility is fitted with its previous values.

**auxiliary regression**: a second stage regression that is usually not of direct interest in its own right, but rather is conducted in order to test the statistical adequacy of the original regression model.

## B

**balanced panel**: a dataset where the variables have both time series and cross-sectional dimensions, and where there are equally long samples for each cross-sectional entity (i.e. no missing data).

**Bayes information criterion**: *see *Schwarz’s Bayesian information criterion (SBIC).

**BDS test**: a test for whether there are patterns in a series, predominantly used for determining whether there is evidence for nonlinearities.

**BEKK model**: a multivariate model for volatilities and covariances between series that ensures the variance–covariance matrix is positive definite.

**BHHH algorithm**: a technique that can be used for solving optimisation problems including maximum likelihood.

**backshift operator**: *see *lag operator.

**Bera–Jarque test**: a widely employed test for determining whether a series closely approximates a normal distribution.

**best linear unbiased estimator (BLUE)**: is one that provides the lowest sampling variance and which is also unbiased.

**between estimator**: is used in the context of a fixed effects panel model, involving running a cross-sectional regression on the time averaged values of all the variables in order to reduce the number of parameters requiring estimation.

**biased estimator**: where the expected value of the parameter to be estimated is not equal to the true value.

**bid–ask spread**: the difference between the amount paid for an asset (the ask or offer price) when it is purchased and the amount received if it is sold (the bid).

**binary choice**: a discrete choice situation with only two possible outcomes.

**bivariate regression**: a regression model where there are only two variables – the dependent variable and a single independent variable.

**bootstrapping**: a technique for constructing standard errors and conducting hypothesis tests that requires no distributional assumptions and works by resampling from the data.

**Box–Jenkins approach**: a methodology for estimating ARMA models.

**Box–Pierce ***Q***-statistic**: a general measure of the extent to which a series is autocorrelated.

**break date**: the date at which a structural change occurs in a time series or in a model’s parameters.

**Breusch–Godfrey test**: a test for autocorrelation of any order in the residuals from an estimated regression model, based on an auxiliary regression of the residuals on the original explanatory variables plus lags of the residuals.

**broken trend**: a process which is a deterministic trend with a structural break.

## C

**calendar effects**: the systematic tendency for a series, especially stock returns, to be higher at certain times than others.

**capital asset pricing model (CAPM)**: a financial model for determining the expected return on stocks as a function of their level of market risk.

**capital market line (CML)**: a straight line showing the risks and returns of all combinations of a risk-free asset and an optimal portfolio of risky assets.

**Carhart model**: a time series model for explaining the performance of mutual funds or trading rules based on four factors: excess market returns, size, value and momentum.

**causality tests**: a way to examine whether one series leads or lags another.

**censored dependent variable**: where values of the dependent variable above or below a certain threshold cannot be observed, while the corresponding values for the independent variables are still available.

**central limit theorem**: the mean of a sample of data having any distribution converges upon a normal distribution as the sample size tends to infinity.

**chaos theory**: an idea taken from the physical sciences whereby although a series may appear completely random to the naked eye or to many statistical tests, in fact there is an entirely deterministic set of non-linear equations driving its behaviour.

**Chow test**: an approach to determine whether a regression model contains a change in behaviour (structural break) part-way through based on splitting the sample into two parts, assuming that the break-date is known.

**Cochrane–Orcutt procedure**: an iterative approach that corrects standard errors for a specific form of autocorrelation.

**coefficient of multiple determination**: *see R*2.

**cointegration**: a concept whereby time series have a fixed relationship in the long run.

**cointegrating vector**: the set of parameters that describes the long-run relationship between two or more time series.

**common factor restrictions**: these are the conditions on the parameter estimates that are implicitly assumed when an iterative procedure such as Cochrane–Orcutt is employed to correct for autocorrelation.

**conditional expectation**: the value of a random variable that is expected for time *t *+ *s *(*s *= 1*, *2*, . . .*) given information available until time *t*.

**conditional mean**: the mean of a series at a point in time *t *fitted given all information available until the previous point in time *t *− 1.

**conditional variance**: the variance of a series at a point in time *t *fitted given all information available until the previous point in time *t *− 1.

**confidence interval**: a range of values within which we are confident to a given degree (e.g. 95% confident) that the true value of a given parameter lies.

**confidence level**: one minus the significance level (expressed as a proportion rather than a percentage) for a hypothesis test.

**consistency**: the desirable property of an estimator whereby the calculated value of a parameter converges upon the true value as the sample size increases.

**contemporaneous terms**: those variables that are measured at the same time as the dependent variable – i.e. both are at time *t*.

**continuous variable**: a random variable that can take on any value (possibly within a given range).

**convergence criterion**: a pre-specified rule that tells an optimiser when to stop looking further for a solution and to stick with the best one it has already found.

**copulas**: a flexible way to link together the distributions for individual series in order to form joint distributions.

**correlation**: a standardised measure, bounded between −1 and +1, of the strength of association between two variables.

**correlogram**: *see *autocorrelation function.

**cost of carry (COC) model**: shows the equilibrium relationship between spot and corresponding futures prices where the spot price is adjusted for the cost of ‘carrying’ the spot asset forward to the maturity date.

**covariance matrix**: *see *variance–covariance matrix.

**covariance stationary process**: *see *weakly stationary process.

**covered interest parity (CIP)**: states that exchange rates should adjust so that borrowing funds in one currency and investing them in another would not be expected to earn abnormal profits.

**credit rating**: an evaluation made by a ratings agency of the ability of a borrower to meet its obligations to meet interest costs and to make capital repayments when due.

**critical values (CV)**: key points in a statistical distribution that determine whether, given a calculated value of a test statistic, the null hypothesis will be rejected or not.

**cross-equation restrictions**: a set of restrictions needed for a hypothesis test that involves more than one equation within a system.

**cross-sectional regression**: a regression involving series that are measured only at a single point in time but across many entities.

**cumulative distribution**: a function giving the probability that a random variable will take on a value lower than some pre-specified value.

**CUSUM and CUSUMSQ tests**: tests for parameter stability in an estimated model based on the cumulative sum of residuals (CUSUM) or cumulative sum of squared residuals (CUSUMSQ) from a recursive regression.

## D

**daily range estimator**: a crude measure of volatility calculated as the difference between the day’s lowest and highest observed prices.

**damped sine wave**: a pattern, especially in an autocorrelation function plot, where the values cycle from positive to negative in a declining manner as the lag length increases.

**data generating process (DGP)**: the true relationship between the series in a model.

**data mining**: looking very intensively for patterns in data and relationships between series without recourse to financial theory, possibly leading to spurious findings.

**data revisions**: changes to series, especially macroeconomic variables, that are made after they are first published.

**data snooping**: *see *data mining.

**day-of-the-week effect**: the systematic tendency for stock returns to be higher on some days of the week than others.

**degrees of freedom**: a parameter that affects the shape of a statistical distribution and therefore its critical values. Some distributions have one degree of freedom parameter, while others have more.

**degree of persistence**: the extent to which a series is positively related to its previous values.

**dependent variable**: the variable, usually denoted by *y *that the model tries to explain.

**deterministic**: a process that has no random (stochastic) component.

**Dickey–Fuller (DF) test**: an approach to determining whether a series contains a unit root, based on a regression of the change in that variable on the lag of the level of that variable.

**differencing**: a technique used to remove a (stochastic) trend from a series that involves forming a new series by taking the lagged value of the original series away from the current one.

**differentiation**: a mathematical technique to find the derivative, which is the slope of a function, or in other words the rate at which *y *changes in response to changes in *x*.

**discrete choice**: a model where the key variable takes only integer values that capture the selections made between alternatives – for example, between modes of transport for a particular journey.

**discrete variable**: a random variable that can only take specific values.

**distributed lag models**: contain lags of the explanatory variables but no lags of the explained variable.

**disturbance term**: *see *error term.

**double logarithmic form**: a specification of a model where logarithms are taken of both the dependent variable (*y*) and the independent variable(s) (*x*).

**dummy variables**: artificially constructed variables that capture qualitative information – for example, for male/female, days of the week, emerging/developed markets, etc. They are usually binary variables (0 or 1).

**Durbin–Watson (DW) statistic**: a test for first order autocorrelation, i.e. a test for whether a (residual) series is related to its immediately preceding values.

**dynamic conditional correlation**: a model that explicitly models correlations in a timevarying, autoregressive fashion.

**dynamic model**: a model that includes lagged or differenced terms of the dependent or independent variables (or both).

## E

**efficient estimator**: an approach to parameter estimation that is optimal in some sense. In econometrics, this is usually taken to mean a formula for calculating the parameters that leads to minimum sampling variance; in other words, the estimates vary as little as possible from one sample to another.

**efficient frontier**: a curve that traces out all possible optimal portfolios.

**efficient market hypothesis**: the notion that asset prices will rapidly reflect all relevant and available information.

**eigenvalues**: the characteristic roots of a matrix.

**eigenvectors**: a set of vectors that, when multiplied by a square matrix, give a set of vectors that differ from the originals by a multiplicative scalar.

**elasticities**: the responsiveness of a percentage change in one variable to percentage changes in another.

**encompassing principle**: the notion that a good model will be able to explain all that competing models can and more.

**encompassing regression**: a hybrid model that incorporates the variables contained in two or more competing models as a method of selecting which is the best between them. The parameters of the best model will be significant in the hybrid model.

**endogenous variable**: a variable whose value is determined within the system of equations under study. In the context of a simultaneous system, each endogenous variable has its own equation specifying how it is generated.

**Engle–Granger (EG) test**: a unit root test applied to the residuals of a potentially cointegrating regression.

**Engle–Ng test**: a test for appropriate specification of a GARCH model in terms of whether there are any uncaptured asymmetries.

**equilibrium correction model**: *see *error correction model.

**error correction model (ECM)**: a model constructed using variables that are employed in stationary, first-differenced forms together with a term that captures movements back towards long run equilibrium.

**error term**: part of a regression model that sweeps up any influences on the dependent variable that are not captured by the independent variables.

**errors-in-variables regression**: a valid approach to estimating the parameters of a regression when the explanatory variables are measured with error and are thus stochastic.

**estimate**: the calculated value of a parameter obtained from the sample data.

**estimator**: an equation that is employed together with the data in order to calculate the parameters that describe the regression relationship.

**exogeneity**: the extent to which a variable is determined outside of the model under study.

**event study**: an approach to financial research where the impact of an identifiable event (e.g. a dividend announcement) is measured on a firm characteristic (e.g. its stock price) to evaluate the market reaction to the event.

**exogenous variables**: variables whose values are taken as given and are determined outside of the equation or system of equations under study and are thus not correlated with the error term.

**expectations hypothesis**: related particularly to the term structure of interest rates. It states that the expected return from investing in a long termbond will be equal to the return from investing in a series of short-term bonds plus a risk premium. In other words, the long term interest rate is a geometric average of the current and expected future short term rates (plus a risk premium).

**explained sum of squares (ESS)**: the part of the variation in *y *that is explained by the model.

**explained variable**: *see *dependent variable.

**explanatory variables**: those variables which are on the right hand side of an equation, whose values are usually taken as fixed, and which are purported to be explaining the values of the dependent variable *y*.

**exponential (EGARCH)**: a model where volatility is modelled in an exponential form so that no non-negativity conditions need to be applied to the parameters. This specification also that allows for asymmetries in the relationship between volatility and returns of different signs.

**exponential growth model**: a model where the dependent variable is an exponential function of one or more independent variables.

**exponential smoothing**: a simple approach to modelling and forecasting where the current smoothed value is a geometrically declining function of all previous values of the series.

**exponentially weighted moving average (EWMA) model**: a simple method for modelling and forecasting volatility where the current estimate is simply a weighted combination of previous values, with the weightings exponentially declining back through time.

## F

*F***-statistic**: a measure that follows an *F*distribution used for testingmultiple hypotheses.

**factor loading**: has several meanings but in particular in the context of principal component analysis, it gives the amount of a variable that appears in each component.

**Fama–MacBeth procedure**: a two-step procedure for testing asset pricing models such as the CAPM. In the first stage the betas are estimated in a set of time series regressions and then a second stage cross-sectional regression examines the explanatory power of these betas.

**financial options**: securities that give the holder the right but not the obligation to buy or sell another asset at a pre-specified price on a pre-specified date.

**first differences**: new series constructed by taking the immediately previous value of a series from its current value.

**fitted value**: the value of *y *that the model fits for a given data point, i.e. for given values of the explanatory variable.

**fixed effects**: most commonly a type of model used for panel data that employs dummies to account for variables that affect the dependent variable *y *cross-sectionally but do not vary over time. Alternatively, the dummies can capture variables that affect *y *over time but do not vary cross-sectionally.

**forcing variable**: sometimes used synonymously with explanatory variable; alternatively it can mean the unobservable statedetermining variable that governs the regime in a Markov switching regression model.

**forecast encompassing test**: a regression of the actual values of a series on several corresponding sets of forecasts. The idea is that if a parameter estimate is statistically significant, then the forecasts from the corresponding model encompass (i.e. contain more information than) those of the other model(s).

**forecast error**: the difference between the actual value of a series and the value that has been forecast for it.

**forward rate unbiasedness (FRU)**: the hypothesis that the forward rate of foreign exchange should be an unbiased prediction of the future spot rate of interest.

**fractionally integrated models**: a way to represent series that are stationary but highly persistent and thus have long memory.

**functional form misspecification**: *see *RESET test.

**futures prices**: the price of a specific quantity of a good or asset for delivery at some pre-specified date in the future.

## G

**GARCH-in-mean (GARCH-M)**: a dynamic model for volatility where the standard deviation (or variance) enters into the generating process for returns.

**Gauss–Markov theorem**: a derivation using algebra showing that, providing a certain set of assumptions hold, the OLS estimator is the best linear unbiased estimator (BLUE).

**general-to-specific methodology**: a philosophical approach to constructing econometric models where the researcher commences with a very broad model and then, through hypothesis testing, reduces the model down to a smaller one.

**generalised autoregressive conditional heteroscedasticity (GARCH) models**: a common specification of dynamic model for volatility.

**generalised least squares (GLS)**: an approach to the estimation of econometric models that is more flexible than ordinary least squares and can be used to relax one or more of its limiting assumptions.

**generalised unrestricted model (GUM)**: the initial, broad model that is specified as the first step of the general-to-specific approach to model construction.

**gilt–equity yield ratio (GEYR)**: the ratio of the yield on long term Treasury bonds to the dividend yield on stocks.

**GJR model**: a model for time-varying volatilities developed by Glosten, Jaganathan and Runkle to allow for asymmetries in the relationship between volatility and returns of different signs.

**Goldfeld–Quandt test for heteroscedasticity**: one of several available tests for whether the residuals froman estimated model have constant variance.

**goodness of fit statistic**: a measure of how well the model that has been estimated fits the sample data.

**Granger representation theorem**: states that if there exists a dynamic linear model with stationary disturbances but where the component variables are non-stationary, then they must be cointegrated.

## H

**Hamilton’s filter**: a form of Markovswitching model where an unobservable state variable switches between discrete regimes via a first-order Markov process.

**Hannan–Quinn information criterion**: a metric that can be used to select the best fitting from a set of competing models and that incorporates a moderate penalty term for including additional parameters.

**Hausman test**: a test for whether a variable can be treated as exogenous or whether in fact the researcher needs to specify a separate structural equation for that variable. It can also refer to a test for whether a random effects approach to panel regression is valid or whether a fixed effects model is necessary.

**Heckman procedure**: a two-step method that corrects for the selection bias that can be observed in the context of samples not selected randomly.

**hedge ratios**: in the context of hedging with futures contracts, this is the number of futures contracts that are sold per unit of the spot asset held.

**hedonic pricing models**: a modelling approach where the price of a physical asset is modelled as a function of its characteristics.

**heteroscedasticity**: where the variance of a series is not constant throughout the sample.

**heteroscedasticity-robust**: a set of standard errors (or test statistics) that have been calculated using an approach that is valid in the presence of heteroscedastic residuals.

**hypothesis test**: a framework for considering plausible values of the true population parameters given the sample estimates.

## I

**identification**: a condition for whether all of the structural parameters in a particular equation from a simultaneous system can be retrieved from estimating the corresponding reduced form equation.

**identity matrix**: a square matrix containing ones on the main diagonal and zeros everywhere else.

**implied volatility models**: an approach whereby the volatility of an underlying asset is calculated from the traded price of an option and a pricing formula.

**impulse responses**: an examination of the impact of a unit shock to one variable on the other variables in a vector autoregressive (VAR) system.

**independent variables**: *see *explanatory variables.

**information criteria**: a family of methods for selecting between competing models that incorporate automatic correction penalties when larger numbers of parameters are included.

**instrumental variables (instruments)**: can be used to replace endogenous variables on the right hand side of a regression equation. The instruments are correlated with the variables they replace but not with the error term in the regression.

**integrated GARCH (IGARCH)**: a model where the variance process is non-stationary so that the impact of shocks on volatility persists indefinitely.

**integrated variable**: one which requires differencing to make it stationary.

**interactive dummy variable**: when a dummy variable is multiplied by an explanatory variable to allow the regression slope to change according to the value of the dummy.

**intercept**: the point where a regression line crosses the *y*-axis, also known sometimes as ‘the coefficient on the constant term’, or sometimes just ‘the constant term’.

**inverse (of a matrix)**: a transformed matrix which, when multiplied by the original matrix, yields the identity matrix.

**invertibility**: a condition for a moving average (MA) model to be representable as a valid infinite-order autoregressive model.

**irrelevant variables**: variables that are included in a regression equation but in fact have no impact on the dependent variable.

## J

**Jensen’s alpha**: the intercept estimate in a regression model of the returns to a portfolio or strategy on a risk factor or set of risk factors, especially in the context of the CAPM. Alpha measures the degree to which there was abnormally bad or good performance.

**Johansen test**: an approach to determining whether a set of variables is cointegrated – i.e. if they have a long-run equilibrium relationship.

**joint hypothesis**: a multiple hypothesis that involves making more than one restriction simultaneously.

**just identified equation**: occurs when the parameters in a structural equation from a system can be uniquely obtained by substitution from the reduced form estimates.

## K

**KPSS test**: a test for stationarity – in other words, a test where the null hypothesis is that a series is stationary against an alternative hypothesis that it is not.

**kurtosis**: the standardised fourth moment of a series; a measure of whether a series has ‘fat tails’.

## L

**lag length**: the number of lagged values of a series used in a model.

**lag operator**: an algebraic notation for taking the current value of a series and turning it into a past value of that series.

**Lagrange multiplier (LM) test**: used in the context of maximum-likelihood estimation, an LM test involves estimation of a restricted regression only. In practice, an LM test is often employed via the calculation of *R*2 from an auxiliary regression to construct a test statistic that follows a *χ*2 distribution.

**law of large numbers**: a theorem stating that the mean from a sample will approach the true population mean (i.e. the expected value) as the sample size increases.

**least squares**: *see *ordinary least squares.

**least squares dummy variables (LSDV)**: an approach to estimating panel data models using 0–1 intercept dummy variables for each cross-sectional unit.

**leptokurtosis**: a phenomenon whereby a series has a higher peak at the mean and fatter tails than a normal distribution with the same mean and variance.

**leverage effects**: the tendency for stock volatility to rise more following a large stock price fall than a price rise of the same magnitude owing to the consequent impact on the firm’s debt-to-equity (leverage) ratio.

**likelihood function**: a mathematical expression that relates to the data and the parameters. A likelihood function is constructed given an assumption about the distribution of the errors, and then the values of the parameters that maximise it are chosen.

**likelihood ratio (LR) test**: an approach to hypothesis testing arising from maximum likelihood estimation that revolves around a comparison of the maximised values of the log-likelihood functions for the restricted and unrestricted models.

**limited dependent variable**: when the values that the dependent variable can take are restricted in some way. In such cases, OLS cannot be validly used to estimate the model parameters.

**linear probability model**: a simple but flawed model for use when the dependent variable in a regression model is binary (0 or 1).

**linearity**: the extent to which a relationship between variables can be represented by a (possibly multi-dimensional) straight line.

**Ljung–Box test**: a general test for autocorrelation in a variable or residual series.

**log-likelihood function (LLF)**: the natural logarithm of the likelihood function.

**log-log model**: *see *double logarithmic form.

**logit model**: an approach for use when the dependent variable in a regression model is binary (0 or 1), and which ensures that the estimated probabilities are bounded by 0 and 1.

**long-memory models**: *see *fractionally integrated models.

**long-run static solution**: the algebraic manipulation of a dynamic equation to construct the long-run relationship between the variables.

**longitudinal data**: *see *panel data.

**loss function**: is constructed in order to evaluate the accuracy of a model fit or of forecasts. The parameters of a model are usually estimated by minimising or maximising a loss function.

**Lyapunov exponent**: a characteristic that can be used to determine whether a series can be described as chaotic.

## M

**marginal effects**: the impacts of changes in the explanatory variables on changes in the probabilities for probit and logit models. They are calculated in order to intuitively interpret the models.

**marginal probability**: the probability of a single random variable.

**market microstructure**: a financial term, concerned with the way that markets work and the impact that the design and structure of the market can have on the outcomes of trade, including prices, volumes and execution costs.

**market risk premium**: the amount of additional return that an investor requires for accepting an additional unit of market risk, often calculated as the difference between the returns on a broad portfolio of stocks and a proxy for the risk free rate of interest.

**market timing**: the extent to which investors are able to select the optimal times to invest in different asset classes.

**Markov switching model**: a time series approach based on a dependent variable that alternates between regimes according to the value of an unobservable state variable that follows a Markov process.

**Marquardt algorithm**: an approach to optimisation that can be used, for example, as part of the procedure to estimate the parameter values in maximum likelihood estimation.

**matrix**: a two-dimensional array of numbers constructed in rows and columns.

**maximum likelihood (ML)**: an approach that can be used for parameter estimation based on the construction and maximisation of a likelihood function, which is particularly useful for non-linear models.

**minimum capital risk requirement (MCRR)**: *see *value-at-risk.

**misspecification error**: occurs when the model estimated is incorrect – for example, if the true relationship between the variables is non-linear but a linear model is adopted.

**misspecification tests**: are diagnostic tests that can provide the researcher with information concerning whether a model has desirable statistical properties, particularly regarding the residuals.

**model interpretation**: the examination of an estimated model in terms of whether the signs of the parameters (i.e. positive or negative) and sizes of the parameters (i.e. their values) make sense intuitively.

**moments**: the moments of a distribution describe its shape. The first moment of a distribution is the mean, the second moment is the variance, the third (standardised) moment is the skewness and the fourth (standardised) moment is the kurtosis. The fifth moments and higher are harder to interpret and in general are not calculated.

**moving average (MA) process**: a model where the dependent variable depends upon the current and past values of a white noise (error) process.

**multicollinearity**: a phenomenon where two or more of the explanatory variables used in a regression model are highly related to one another.

**multimodal**: a characteristic of a distribution whereby it does not have a single peak at the mean, but rather reaches a maximum in more than one place.

**multinomial logit or probit**: classes of models that are used for discrete choice problems, where we wish to explain how individuals make choices between more than two alternatives.

**multivariate generalised autoregressive conditionally heteroscedastic (GARCH) models**: a family of dynamic models for timevarying variances and covariances.

## N

**neural network models**: a class of statistical models whose structure is loosely based on how computation is performed by the brain. They have been employed for time series modelling and for classification purposes.

**Newey–West estimator**: a procedure that can be employed to adjust standard errors to allow for heteroscedasticity and/or autocorrelation in the residuals from a regression model.

**news impact curve**: a pictorial representation of the responsiveness of volatility to positive and negative shocks of different magnitudes.

**Newton–Raphson procedure**: an iterative approach to optimisation – in other words, for finding the values of a parameter or set of parameters that maximise or minimise a function.

**nominal series**: a series that has not been deflated (i.e. not been adjusted for inflation).

**non-linear least squares (NLS)**: an estimation technique for use on non-linear models (models that are non-linear in the parameters) based on minimising the sum of the squared residuals.

**non-negativity constraints**: the conditions that it is sometimes necessary to impose on the parameter estimates from non-linear models to ensure that they are not negative in situations where it would not make sense for them to be so.

**non-nested models**: where there are at least two models, neither of which is a special (i.e. restricted) case of the other.

**non-normality**: not following a normal or Gaussian distribution.

**non-stationarity**: a characteristic of a time series whereby it does not have a constant mean, a constant variance, and a constant autocovariance structure.

**null hypothesis**: a formal expression of the statement actually being tested as part of a hypothesis test.

## O

**observations**: another name for the data points available for analysis. **omitted variable**: a relevant variable for explaining the dependent variable has been left out of the estimated regression equation, leading to biased inferences on the remaining parameters.

**one-sided hypothesis test**: used when theory suggests that the alternative hypothesis should be of the greater than form only or of the less than form only (and not both).

**optimal portfolio**: a combination of risky assets that maximises return for a given risk or minimises risk for a given return.

**order of integration**: the number of times that a stochastically non-stationary series must be differenced to make it stationary.

**ordered response variable**: usually a situation where the dependent variable in a model is limited to only certain values but where there is a natural ordering of those values – for example, where the values represent sovereign credit rating assignments.

**ordinal scale**: where a variable is limited so that its values define a position or ordering only, and thus the precise values that the variable takes have no direct interpretation.

**ordinary least squares (OLS)**: the standard and most common approach that is used to estimate linear regression models.

**out-of-sample**: sometimes, not all observations are employed to estimate the model (insample data), but instead some are retained for forecasting (the out-of-sample data).

**outliers**: data points that do not fit in with the pattern of the other observations and that are a long way from the fitted model.

**overfitting**: estimating too large a model with too many parameters.

**overidentified equation**: occurswhen more than one estimate of each parameter in the structural equation from a system can be obtained by substitution from the reduced form estimates.

**overreaction effect**: the tendency for asset (especially stock) prices to overshoot their new equilibrium prices when news is released.

**oversized test**: a statistical test that rejects the null hypothesis too often when it is in fact correct.

## P

*p***-value**: the exact significance level, or the marginal significance level which would make us indifferent between rejecting and not rejecting the null hypothesis.

**panel data analysis**: the use of data having both cross-sectional and time series dimensions.

**parsimonious model**: one that describes the data accurately while using as few parameters as possible.

**partial autocorrelation function (pacf)**: measures the correlation of a variable with its value *k *periods ago (*k *= 1*, *2*, . . .*) after removing the effects of observations at all intermediate lags.

**pecking order hypothesis**: the notion from corporate finance that firms will select the cheapest method of financing (usually retained earnings) first before switching to increasingly more expensive forms.

**perfect multicollinearity**: occurs when an explanatory variable used in a regression model is a precise linear combination of one or more other explanatory variables from that model.

**period effects**: *see *time fixed effects.

**piecewise linear model**: a model that is linear (i.e. can be represented by a straight line) within restricted ranges of the data, but where taken overall the model is non-linear.

**pooled sample**: where there is a panel of data (i.e. having both time series and crosssectional dimensions), but where all of the observations are employed together without regard for the panel structure.

**population**: the collection of all objects or entities that are relevant to the idea being tested in a model.

**population regression function (PRF)**: embodies the true but unobservable relationship between the dependent and independent variables.

**portmanteau tests**: general tests for nonlinear patterns or model-misspecification; in other words, tests that have power over a broad range of alternative structures.

**position risk requirement**: *see *value-atrisk.

**power of a test**: the ability of a test to correctly reject a wrong null hypothesis.

**pre-determined variables**: are uncorrelated with past or current values of the error term in a regression equation but may be correlated with future values of the error term.

**predicted value**: *see *fitted value.

**predictive failure test**: a test for parameter stability or structural change in a regression model, which is based on estimating an auxiliary regression for a sub-sample of the data and then evaluating how well that model can ‘predict’ the other observations.

**price deflator**: a series that measures the general level of prices in an economy, used to adjust a nominal series to a real one.

**principal components analysis (PCA)**: a technique that is sometimes used where a set of variables are highly correlated. More specifically, it is a mathematical operation that converts a set of correlated series into a new set of linearly independent series.

**probability density function (pdf)**: a relationship or mapping that describes how likely it is that a random variable will take on a value within a given range.

**probit model**: an appropriate model for binary (0 or 1) dependent variables where the underlying function used to transform the model is a cumulative normal distribution.

**pseudo-random numbers**: a set of artificial random-looking numbers generated using a purely deterministic sequence (e.g. using a computer).

**purchasing power parity (PPP)**: the hypothesis that, in equilibrium, exchange rates should adjust so that a representative basket of goods and services should cost the same when converted into a common currency irrespective of where it was purchased.

## Q

**qualitative variables**: *see *dummy variables.

**Quandt likelihood ratio test**: a test for structural breaks in a regression model, based on the Chow test but where the break date is assumed unknown.

**quantile**: the position (within the 0–1 interval) in an ordered series where an observation falls.

**quantile regression**: an approach to model specification that involves constructing a family of regression models, each for different quantiles of the distribution of the dependent variable.

## R

** R**2: a standardised measure, bounded between zero and one, of how well a sample regression model fits the data.

*R***-bar**2: *see *adjusted *R*2.

**random effects model**: a particular type of panel data model specification where the intercepts vary cross-sectionally as a result of each cross-sectional entity having a different error term.

**random walk**: a simple model where the current value of a series is simply the previous value perturbed by a white noise (error) term. Therefore the optimal forecast for a variable that follows a random walk is the most recently observed value of that series.

**random walk with drift**: a random walk model that also includes an intercept, so that changes in the variable are not required to average zero.

**rank (of a matrix)**: a measure of whether all the rows and columns of a matrix are independent of one another.

**real series**: a series that has been deflated (adjusted for inflation).

**recursive model**: an approach to estimation where a set of time series regressions are estimated using sub-samples of increasing length. After the first model is estimated, an additional observation is added to the end of the sample so that the sample size increases by one observation. This continues until the end of the sample is reached.

**reduced form equations**: the equations with no endogenous variables on the righthand side that have been derived algebraically from the structural forms in the context of a simultaneous system.

**redundant fixed effects test**: a test for whether a fixed effects panel regression approach must be employed, or whether the data can simply be pooled and estimated using a standard ordinary least squares regression model.

**regressand**: *see *dependent variable.

**regressor**: *see *explanatory variable.

**rejection region**: if a test statistic falls within this area plotted onto a statistical distribution function then the null hypothesis under study is rejected.

**re-sampling**: creating a simulated distribution for computing standard errors or critical values via sampling with replacement from the original data.

**RESET test**: a non-linearity test, or a test for misspecification of functional form, i.e. a situation where the shape of the regression model estimated is incorrect – for example, where the model estimated is linear but it should have been non-linear.

**residual diagnostics**: an examination of the residuals for whether they have any patterns remaining that were present in the dependent variable and not captured by the fitted model.

**residual sum of squares (RSS)**: the addition of all of the squared values of the differences between the actual data points and the corresponding model fitted values.

**residual terms**: the differences between the actual values of the dependent variable and the values that the model estimated for them – in other words, the parts of the dependent variable that the model could not explain.

**restricted model**: a regression where the parameters cannot be freely determined by the data, but instead some restrictions have been placed on the values that can be taken by one or more of the parameters.

**risk premium**: the additional return that investors expect for bearing risk.

**riskless arbitrage opportunities**: *see *arbitrage.

**rolling window**: an approach to estimation where a set of time series regressions are estimated using sub-samples of fixed length. After the first model is estimated, the first observation is removed from the sample and one observation is added to the end. This continues until the end of the sample is reached.

## S

**sample**: a selection of some entities from the population which are then used to estimate a model.

**sample regression function (SRF)**: the regression model that has been estimated from the actual data.

**sample size**: the number of observations or data points per series in the sample.

**sampling error**: the inaccuracy in parameter estimation that arises as a result of having only a sample and not the whole population; as a consequence of sampling error, the estimates vary from one sample to another.

**Schwarz’s Bayesian information criterion (SBIC)**: a metric that can be used to select the best fitting from a set of competing models and that incorporates a strict penalty term for including additional parameters.

**second moment**: the moments of a distribution define its shape; the second moment is another term for the variance of the data.

**seemingly unrelated regression (SUR)**: a time series regression approach for modelling the movements of several highly related dependent variables. The approach allows for the correlation between the error terms of the regressions, hence improving the efficiency of the estimation.

**self-exciting threshold autoregression (SETAR)**: a TAR model where the state determining variable is the same as the variable under study.

**semi-interquartile range**: a measure of the spread of a set of data (an alternative to the variance) that is based on the difference between the quarter- and three-quarter points of the ordered data.

**sensitive dependence on initial conditions (SDIC)**: this is the defining characteristic of a chaotic system that the impact on a system of an infinitesimally small change in the initial values will grow exponentially over time.

**serial correlation**: *see *autocorrelation.

**Sharpe ratio**: in finance, this is a risk adjusted performance measure calculated by subtracting the risk-free return from the portfolio return, and then dividing this by the portfolio standard deviation.

**shocks**: another name for the disturbances in a regression model.

**short-selling**: selling a financial asset that you do not own, in anticipation of repurchasing it at a later date when the price has fallen.

**significance level**: the size of the rejection region for a statistical test, also equal to the probability that the null hypothesis will be rejected when it is correct.

**sign and size bias tests**: tests for asymmetries in volatility – i.e. tests for whether positive and negative shocks of a given size have the same effect on volatility.

**simultaneous equations**: a set of interlinked equations each comprising several variables.

**size of test**: *see *significance level.

**skewness**: the standardised third moment of a distribution that shows whether it is symmetrical around its mean value.

**slippage time**: the amount of time that it is assumed to take to execute a trade after a rule is computer-generated.

**slope**: the gradient of a straight (regression) line, measured by taking the change in the dependent variable, *y *between two points, and dividing it by the change in the independent variable, *x *between the same points.

**sovereign credit ratings**: are assessments of the riskiness of debts issued by governments.

**sovereign yield spreads**: usually defined as the difference between the yield on the bonds of a government under study and the yield on US Treasury bonds.

**specific-to-general modelling**: a philosophical approach to building econometric models that involves starting with a specific model as indicated by theory and then sequentially adding to it or modifying it so that it gradually becomes a better description of reality.

**spline techniques**: piecewise linear models that involve the application of polynomial functions in a piecewise fashion to different portions of the data.

**spot price**: the price of a specific quantity of a good or asset for immediate delivery.

**spurious regressions**: if a regression involves two or more independent non-stationary variables, the slope estimate(s) may appear highly significant to standard statistical tests and may have highly significant *t*-ratios even though in reality there is no relationship between the variables.

**standard deviation**: a measure of the spread of the data about their mean value, which has the same units as the data.

**standard errors**: measure the precision or reliability of the regression estimates.

**stationary variable**: one that does not contain a unit or explosive root and can thus be validly used directly in a regression model.

**statistical inference**: the process of drawing conclusions about the likely characteristics of the population from the sample estimates.

**statistically significant**: a result is statistically significant if the null hypothesis is rejected (usually using a 5% significance level).

**stochastic regressors**: it is usually assumed when using regression models that the regressors are non-stochastic or fixed; in practice, however, they may be random or stochastic – for example, if there are lagged dependent variables or endogenous regressors.

**stochastic trend**: some levels time series possess a stochastic trend, meaning that they can be characterised as unit root processes, which are non-stationary.

**stochastic volatility (SV) model**: a less common alternative to GARCH models, where the conditional variance is explicitly modelled using an equation containing an error term.

**strictly exogenous variable**: one that is uncorrelated with past, present and future values of the error term.

**strictly stationary process**: one where the entire probability distribution is constant over time.

**structural break**: a situation where the properties of a time series or of a model exhibit a substantial long-term shift in behaviour.

**structural equations**: the original equations describing a simultaneous system, which contain endogenous variables on the right hand side.

**sum of squared residuals**: *see *residual sum of squares.

**switching model**: an econometric specification for a variable whose behaviour alternates between two or more different states.

## T

*t***-ratio**: the ratio of a parameter estimate to its standard error, forming a statistic to test the null hypothesis that the true value of the parameter is one.

**Theil’s ***U***-statistic**: a metric to evaluate forecasts, where the mean squared error of the forecasts from the model under study is divided by the mean squared error of the forecasts from a benchmark model. A *U*-statistic of less than one implies that the model is superior to the benchmark.

**threshold autoregressive (TAR) models**: a class of time series models where the series under study switches between different types of autoregressive dynamics when an underlying (observable) variable exceeds a certain threshold.

**time fixed effects**: a panel data model that allows the regression intercept to vary over time and is useful when the average value of the variable under study changes over time but not cross-sectionally.

**time series regressions**: models built using time series data – i.e. data collected for a period of time for one or more variables.

**tobit regression**: a model that is appropriate when the dependent variable is censored – that is, where the values of the variable beyond a specific threshold cannot be observed, even though the corresponding values of the independent variables are observable.

**total sum of squares (TSS)**: the sum of the squared deviations of the dependent variable about its mean value.

**transition probabilities**: a square matrix of estimates of the likelihood that a Markov switching variable will move from a given regime to each other regime.

**truncated dependent variable**: a situation where the values of this variable beyond a certain threshold cannot be observed, and neither can the corresponding values of the independent variables.

**two-stage least squares (TSLS or 2SLS)**: an approach to parameter estimation that is valid for use on simultaneous equations systems.

## U

**unbalanced panel**: a set of data having both time series and cross-sectional elements, but where some data are missing – i.e. where the number of time series observations available is not the same for all cross-sectional entities.

**unbiased estimator**: a formula or set of formulae that, when applied, will give estimates that are on average equal to the corresponding true population parameter values.

**uncovered interest parity (UIP)**: holds if covered interest parity and forward rate unbiasedness both apply.

**underidentified or unidentified equation**: occurs when estimates of the parameters in the structural equation from a system cannot be obtained by substitution from the reduced form estimates as there is insufficient information in the latter.

**unit root process**: a series follows a unit root process if it is non-stationary but becomes stationary by taking first differences.

**unparameterised**: if a feature of the dependent variable *y *is not captured by the model, it is unparameterised.

**unrestricted regression**: a model that is specified without any restrictions being imposed so that the estimation technique can freely determine the parameter estimates.

## V

**value-at-risk (VaR)**: an approach to measuring risk based on the loss on a portfolio that may be expected to occur with a given probability over a specific horizon.

**variance–covariance matrix**: an array of numbers that comprises each of the variances of a set of random variables on the leading diagonal of the matrix and their covariances as the off-diagonal elements.

**variance decomposition**: a way to examine the importance of each variable in a vector autoregressive (VAR) model by calculating how much of the forecast error variance (for 1, 2, . . . , periods ahead) for each dependent variable can be explained by innovations in each independent variable.

**variance reduction techniques**: are employed in the context of Monte Carlo simulations in order to reduce the number of replications required to achieve a given level of standard errors of the estimates.

**VECH model**: a relatively simple multivariate approach that allows for the estimation of time-varying volatilities and covariances that are stacked into a vector.

**vector autoregressive (VAR) model**: a multivariate time series specification where lagged values of (all) the variables appear on the right hand side in (all) the equations of the (unrestricted) model.

**vector autoregressive moving average (VARMA) model**: a VAR model where there are also lagged values of the error terms appearing in each equation.

**vector error correction model (VECM)**: an error correction model that is embedded into a VAR framework so that the short- and long-run relationships between a set of variables can be modelled simultaneously.

**vector moving average (VMA) model**: a multivariate time series model where a series is expressed as a combination of lagged values of a vector of white noise processes.

**volatility**: the extent to which a series is highly variable over time, usually measured by its standard deviation or variance.

**volatility clustering**: the tendency for the variability of asset returns to occur ‘in bunches’, so that there are prolonged periods when volatility is high and other prolonged periods when it is low.

## W

**Wald test**: an approach to testing hypotheses where estimation is undertaken only under the alternative hypothesis; most common forms of hypothesis tests (e.g. *t*- and *F*-tests) are Wald tests.

**weakly exogenous variables**: *see *predetermined variables.

**weakly stationary process**: has a constant mean, constant variance and constant autocovariances for each given lag.

**weighted least squares (WLS)**: *see *generalised least squares.

**white noise process**: has a fixed mean and variance but no other structure (e.g. it has zero autocorrelations for all lags). The error term in a regression model is usually assumed to be white noise.

**White’s correction**: an adjustment to the standard errors of regression parameters that allows for heteroscedasticity in the residuals from the estimated equation.

**White’s test**: an approach to determining whether the assumption of homoscedastic errors in a model is valid, based on estimating an auxiliary regression of the squared residuals on the regressors, their squared values, and their cross-products.

**within transformation**: used in the context of a fixed effects panel model, involving the subtraction of the time series mean from each variable to reduce the number of dummy variable parameters requiring estimation.

**Wold’s decomposition theorem**: states that any stationary series can be decomposed into the sum of two unrelated processes, a purely deterministic part and a purely stochastic part.

## Y

** yield curves: **show how the yields on bonds vary as the term to maturity increases.

**Yule–Walker equations**: a set of formulae that can be used to calculate the autocorrelation function coefficients for an autoregressive model.