Notes12 - Lecture Notes 12 Econ 410 Introduction to...

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Lecture Notes 12 Econ 410 – Introduction to Econometrics 1 Time Series Analysis Time series data is data collected for a single entity at multiple points in time. Time series data is mainly used to: 1. analyze the changes over time in the relationship between two or more variables; 2. forecast the future value of some variables of interest. Some definitions: the unit of time, or “period”, can be years, quarters, months, days…. t Y is the value of the variable Y at time t ; 1 t Y is the value of Y in the previous period (called first lagged value or first lag); j t Y is the value of Y j periods ago (the th j lagged value); 1 + t Y is the value of Y one period into the future; 1 = t t t Y Y Y is the first difference in the variable t Y () () ( ) 1 ln ln ln = t t t Y Y Y is the first difference of the logarithm of t Y . It follows that () t t t Y Y Y ln , where the approximation is most accurate when the percentage change is small. Autocorrelation and autocovariance The correlation of a variable with its lagged values is called autocorrelation (or serial correlation). The first autocorrelation is the correlation between t Y and 1 t Y ; in general, the th j autocorrelation is the correlation between t Y and j t Y . In the same way, the first autocovariance is the covariance between t Y and 1 t Y ; in general, the th j autocovariance is the covariance between t Y and j t Y . The th j population autocovariance ( ) j t t Y Y , cov and autocorrelation ( ) j t t j Y Y corr = , ρ are defined by the standard covariance and correlation formulas and can be estimated by their sample counterparts: j T j t T j t T j t j t t Y Y Y Y T Y Y + = + = , 1 1 , 1 1 , cov t j t t j Y Y Y var , cov ˆ =
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Lecture Notes 12 Econ 410 – Introduction to Econometrics 2 In these formulas, T j Y , 1 + is the sample average of t Y computed over the observations T j t ,.... , 1 + = and T is the total number of observations. Note that the formula for the sample autocorrelation assumes that () t Y var i s the same a s 1 var t Y . This is an implication of the assumption of stationarity of t Y , which we discuss next. Stationarity Time series analysis uses historical data to estimate historical relationships between variables. If the future is like the past, then these estimated relationships can be used to forecast the future. However, if the future differs from the past, the forecasts made using historical relationships are not reliable. This is the reason why, in order for the forecasts to be reliable, we need the variables of interest to be stationary. The variable t Y is stationary if its probability distribution does not change over time. More formally, a time series t Y is stationary if the joint distribution of T s s s Y Y Y + + + ,.... , , 2 1 does not depend on s . This condition extends when there are two or more variables. For instance, a pair of time series variables t X and t Y are said to be jointly stationary if the joint distribution of T s T s s s s s Y X Y X Y X + + + + + + , ,.... , , , , 2 2 1 1
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This note was uploaded on 07/28/2010 for the course ECON 410 taught by Professor Staff during the Fall '08 term at University of Wisconsin.

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Notes12 - Lecture Notes 12 Econ 410 Introduction to...

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