Time Series Data

Slide 1: 

Time-Series Data Analysis

Types of Data : 

Types of Data Data collected over a period of time on one or more variables. Data associated with a particular frequency of observation (daily, monthly or annually…) or collection of data points.

Slide 3: 

The Analytical Procedure

The Analytical Procedure : 

The Analytical Procedure Summary Statistics of Data Linear Model Nonlinear Model Luukkonen et al. (1988) Linearity Test If reject not reject Economic or Financial Theory

The Analytical Procedure : 

Time Series Data Unit Root Test Non-Stationarity Staionaruty Orders of Integration The Analytical Procedure Cointegration Test

Slide 6: 

Model Specification The Analytical Procedure Cointegration Test No Yes Unit Root Test Staionaruty EG,JJ

Slide 7: 

Model Estimation The Analytical Procedure

Econometric Soft Packages : 

Econometric Soft Packages

Testing the Normality Assumption : 

Testing the Normality Assumption Why did we need to assume normality for hypothesis testing? Testing for Departures from Normality The Jarque-Bera normality test A normal distribution is not skewed and is defined to have a coefficient of kurtosis of 3. The kurtosis of the normal distribution is 3 so its excess kurtosis (b2-3) is zero.

Normal versus Skewed Distributions : 

Normal versus Skewed Distributions A normal distribution A skewed distribution

Leptokurtic versus Normal Distribution : 

Leptokurtic versus Normal Distribution

Testing for Normality : 

Testing for Normality Bera and Jarque formalise this by testing the residuals for normality by testing whether the coefficient of skewness and the coefficient of excess kurtosis are jointly zero. It can be proved that the coefficients of skewness and kurtosis can be expressed respectively as: and The Bera Jarque test statistic is given by We estimate b1 and b2 using the residuals from the OLS regression, .

What do we do if we find evidence of Non-Normality? : 

What do we do if we find evidence of Non-Normality? It is not obvious what we should do! Could use a method which does not assume normality, but difficult and what are its properties? Often the case that one or two very extreme residuals causes us to reject the normality assumption.

Omission of an Important Variable or Inclusion of an Irrelevant Variable : 

Omission of an Important Variable or Inclusion of an Irrelevant Variable Omission of an Important Variable Consequence: The estimated coefficients on all the other variables will be biased and inconsistent unless the excluded variable is uncorrelated with all the included variables. Even if this condition is satisfied, the estimate of the coefficient on the constant term will be biased. The standard errors will also be biased. Inclusion of an Irrelevant Variable Coefficient estimates will still be consistent and unbiased, but the estimators will be inefficient.

How do we decide the sub-parts to use? : 

How do we decide the sub-parts to use? As a rule of thumb, we could use all or some of the following: - Plot the dependent variable over time and split the data accordingly to any obvious structural changes in the series, e.g. - Split the data according to any known important historical events (e.g. stock market crash, new government elected) - Use all but the last few observations and do a predictive failure test on those.

A Strategy for Building Econometric Models : 

A Strategy for Building Econometric Models The objective to build a statistically adequate empirical model which - satisfies the assumptions of the CLRM - is parsimonious - has the appropriate theoretical interpretation - has the right “shape” - i.e. - all signs on coefficients are “correct” - all sizes of coefficients are “correct” - is capable of explaining the results of all competing models

Approaches to Building Econometric Models : 

Approaches to Building Econometric Models There are 2 popular philosophies of building econometric models: the “specific-to-general” and “general-to-specific” approaches. “Specific-to-general” was used almost universally until the mid 1980’s, and involved starting with the simplest model and gradually adding to it. Little, if any, diagnostic testing was undertaken. But this meant that all inferences were potentially invalid. An alternative and more modern approach to model building is the “LSE” or Hendry “general-to-specific” methodology. The advantages of this approach are that it is statistically sensible and also the theory on which the models are based usually has nothing to say about the lag structure of a model.

The General-to-Specific Approach : 

The General-to-Specific Approach First step is to form a “large” model with lots of variables on the right hand side This is known as a GUM (generalised unrestricted model) At this stage, we want to make sure that the model satisfies all of the assumptions of the CLRM If the assumptions are violated, we need to take appropriate actions to remedy this, e.g. - taking logs - adding lags - dummy variables We need to do this before testing hypotheses Once we have a model which satisfies the assumptions, it could be very big with lots of lags & independent variables

The General-to-Specific Approach: Reparameterising the Model : 

The General-to-Specific Approach: Reparameterising the Model The next stage is to reparameterise the model by - knocking out very insignificant regressors - some coefficients may be insignificantly different from each other, so we can combine them. At each stage, we need to check the assumptions are still OK. Hopefully at this stage, we have a statistically adequate empirical model which we can use for - testing underlying financial theories - forecasting future values of the dependent variable - formulating policies, etc.

Regression Analysis In Practice - A Further Example:Determinants of Sovereign Credit Ratings : 

Regression Analysis In Practice - A Further Example:Determinants of Sovereign Credit Ratings Cantor and Packer (1996) Financial background: What are sovereign credit ratings and why are we interested in them? Two ratings agencies (Moody’s and Standard and Poor’s) provide credit ratings for many governments. Each possible rating is denoted by a grading. The purpose of paper - To attempt to explain and model how the ratings agencies arrived at their ratings. - To use the same factors to explain the spreads of sovereign yields above a risk-free proxy - To determine what factors affect how the sovereign yields react to ratings announcements

Determinants of Sovereign Ratings : 

Determinants of Sovereign Ratings Data Quantifying the ratings (dependent variable): Aaa/AAA=16, ... , B3/B-=1 Explanatory variables (units of measurement): - Per capita income in 1994 (thousands of dollars) - Average annual GDP growth 1991-1994 (%) - Average annual inflation 1992-1994 (%) - Fiscal balance: Average annual government budget surplus as a proportion of GDP 1992-1994 (%) - External balance: Average annual current account surplus as a proportion of GDP 1992-1994 (%) - External debt Foreign currency debt as a proportion of exports 1994 (%) - Dummy for economic development - Dummy for default history Income and inflation are transformed to their logarithms.

What factors are likely to lead to a good forecasting model? : 

What factors are likely to lead to a good forecasting model? “signal” versus “noise” “data mining” issues simple versus complex models financial or economic theory

Limits of forecasting: What can and cannot be forecast? : 

Limits of forecasting: What can and cannot be forecast? All statistical forecasting models are essentially extrapolative Forecasting models are prone to break down around turning points Series subject to structural changes or regime shifts cannot be forecast Predictive accuracy usually declines with forecasting horizon Forecasting is not a substitute for judgement

Back to the original question: why forecast? : 

Back to the original question: why forecast? Why not use “experts” to make judgemental forecasts? Judgemental forecasts bring a different set of problems: e.g., psychologists have found that expert judgements are prone to the following biases: over-confidence inconsistency recency anchoring illusory patterns “group-think”. The Usually Optimal Approach To use a statistical forecasting model built on solid theoretical foundations supplemented by expert judgements and interpretation.

The Test of Significance Approach : 

The Test of Significance Approach 1. We need some tabulated distribution with which to compare the estimated test statistics. Test statistics derived in this way can be shown to follow a t-distribution with T-2 degrees of freedom. As the number of degrees of freedom increases, we need to be less cautious in our approach since we can be more sure that our results are robust. 2. We need to choose a “significance level”, often denoted . This is also sometimes called the size of the test and it determines the region where we will reject or not reject the null hypothesis that we are testing. It is conventional to use a significance level of 5%. Intuitive explanation is that we would only expect a result as extreme as this or more extreme 5% of the time as a consequence of chance alone. Conventional to use a 5% size of test, but 10% and 1% are also commonly used.

Determining the Rejection Region for a Test of Significance : 

Determining the Rejection Region for a Test of Significance 3. Given a significance level, we can determine a rejection region and non-rejection region. For a 2-sided test:

The Rejection Region for a 1-Sided Test (Upper Tail) : 

The Rejection Region for a 1-Sided Test (Upper Tail)

The Rejection Region for a 1-Sided Test (Lower Tail) : 

The Rejection Region for a 1-Sided Test (Lower Tail)

The Test of Significance Approach: Drawing Conclusions : 

The Test of Significance Approach: Drawing Conclusions 4. Use the t-tables to obtain a critical value or values with which to compare the test statistic. 5. Finally perform the test. If the test statistic lies in the rejection region then reject the null hypothesis (H0), else do not reject H0.

A Note on the t and the Normal Distribution : 

A Note on the t and the Normal Distribution You should all be familiar with the normal distribution and its characteristic “bell” shape. We can scale a normal variate to have zero mean and unit variance by subtracting its mean and dividing by its standard deviation. There is, however, a specific relationship between the t- and the standard normal distribution. Both are symmetrical and centred on zero. The t-distribution has another parameter, its degrees of freedom. We will always know this (for the time being from the number of observations -2).

What Does the t-Distribution Look Like? : 

What Does the t-Distribution Look Like?

Testing Multiple Hypotheses: The F-test : 

Testing Multiple Hypotheses: The F-test We used the t-test to test single hypotheses, i.e. hypotheses involving only one coefficient. But what if we want to test more than one coefficient simultaneously? We do this using the F-test. The F-test involves estimating 2 regressions. The unrestricted regression is the one in which the coefficients are freely determined by the data, as we have done before. The restricted regression is the one in which the coefficients are restricted, i.e. the restrictions are imposed on some s.

Calculating the F-Test Statistic : 

Calculating the F-Test Statistic The test statistic is given by where URSS = RSS from unrestricted regression RRSS = RSS from restricted regression m = number of restrictions T = number of observations k = number of regressors in unrestricted regression including a constant in the unrestricted regression (or the total number of parameters to be estimated).

The F-Distribution : 

The F-Distribution The test statistic follows the F-distribution, which has 2 d.f. parameters. The value of the degrees of freedom parameters are m and (T-k) respectively (the order of the d.f. parameters is important). The appropriate critical value will be in column m, row (T-k). The F-distribution has only positive values and is not symmetrical. We therefore only reject the null if the test statistic > critical F-value.

What we Cannot Test with Either an F or a t-test : 

What we Cannot Test with Either an F or a t-test We cannot test using this framework hypotheses which are not linear or which are multiplicative, e.g. H0: 2 3 = 2 or H0: 2 2 = 1 cannot be tested.

The Relationship between the t and the F-Distributions : 

The Relationship between the t and the F-Distributions Any hypothesis which could be tested with a t-test could have been tested using an F-test, but not the other way around. For example, consider the hypothesis H0: 2 = 0.5 H1: 2  0.5 We could have tested this using the usual t-test: or it could be tested in the framework above for the F-test. Note that the two tests always give the same result since the t-distribution is just a special case of the F-distribution. For example, if we have some random variable Z, and Z  t (T-k) then also Z2  F(1,T-k)

Data Mining : 

Data Mining Data mining is searching many series for statistical relationships without theoretical justification. For example, suppose we generate one dependent variable and twenty explanatory variables completely randomly and independently of each other. If we regress the dependent variable separately on each independent variable, on average one slope coefficient will be significant at 5%. If data mining occurs, the true significance level will be greater than the nominal significance level.

Goodness of Fit Statistics : 

Goodness of Fit Statistics We would like some measure of how well our regression model actually fits the data. We have goodness of fit statistics to test this: i.e. how well the sample regression function (srf) fits the data. The most common goodness of fit statistic is known as R2. One way to define R2 is to say that it is the square of the correlation coefficient between y and . For another explanation, recall that what we are interested in doing is explaining the variability of y about its mean value, , i.e. the total sum of squares, TSS: We can split the TSS into two parts, the part which we have explained (known as the explained sum of squares, ESS) and the part which we did not explain using the model (the RSS).

Defining R2 : 

Defining R2 That is, TSS = ESS + RSS Our goodness of fit statistic is But since TSS = ESS + RSS, we can also write R2 must always lie between zero and one. To understand this, consider two extremes RSS = TSS i.e. ESS = 0 so R2 = ESS/TSS = 0 ESS = TSS i.e. RSS = 0 so R2 = ESS/TSS = 1

The Limit Cases: R2 = 0 and R2 = 1 : 

The Limit Cases: R2 = 0 and R2 = 1

Problems with R2 as a Goodness of Fit Measure : 

Problems with R2 as a Goodness of Fit Measure There are a number of them: 1. R2 is defined in terms of variation about the mean of y so that if a model is reparameterised (rearranged) and the dependent variable changes, R2 will change. 2. R2 never falls if more regressors are added. to the regression, e.g. consider: Regression 1: yt = 1 + 2x2t + 3x3t + ut Regression 2: y = 1 + 2x2t + 3x3t + 4x4t + ut R2 will always be at least as high for regression 2 relative to regression 1. 3. R2 quite often takes on values of 0.9 or higher for time series regressions.

Adjusted R2 : 

Adjusted R2 In order to get around these problems, a modification is often made which takes into account the loss of degrees of freedom associated with adding extra variables. This is known as , or adjusted R2: So if we add an extra regressor, k increases and unless R2 increases by a more than offsetting amount, will actually fall. There are still problems with the criterion: 1. A “soft” rule 2. No distribution for or R2