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Common Usage of These Techniques Macroeconomics, finance, time series models Autopilot, radar tracking Orbit tracking, satellite navigation (historically important) Speech, picture enhancement

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Another example Use nightlight data and the Kalman filter to adjust official GDP growth statistics. The idea is that economic activity is closely related to nightlight data. “Measuring Economic Growth from Outer Space” by Henderson, Storeygard, and Weil AER(2012)

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Measuring Long-Term Growth

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Measuring Short-Term Growth

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Measuring Short-Term Growth

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Basic Setup What if you know that are serially correlated: and , Then so one of the assumptions is violated! What to do? Can you still apply the model?

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Kalman Filter: Introduction State Space Representation [univariate case]: Notation: is the best linear predictor of st conditional on the information up to t-1. is the best linear predictor of yt conditional on the information up to t-1. is the best linear predictor of st conditional on the information up to t.

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Kalman Filter: Main Idea Moving from t-1 to t Suppose we know and at time t-1. When arrive in period t we observe and Need to obtain st|t ! If we know , using the state equation: using the observation equation: yt+1|t = axt+1 + bst+1|t The key question: how to obtain st|t from ?

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Kalman Filter: Main Idea How to update st|t ? Idea: use the observed prediction error to infer the state at time t, It turns out it is optimal to update it using is called Kalman gain It measures how informative is the prediction error about the underlying state vector How do you think it depends on the variance of the observation error? It is chosen so that the new prediction error is orthogonal to all of the previous ones. Thus there is no (linear) predictable component in generated errors.

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Kalman Filter: More Notations is the prediction error variance of given the history of observed variables up to t-1. is the prediction error variance of yt conditional on the information up to t-1. is the prediction error variance of conditional on the information up to t. Intuitively the Kalman gain is chosen so that is minimized. Will show this later.

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Kalman Gain: Intuition Kalman gain is chosen so that is minimized. It can be shown that Intuition: If a big mistake is made forecasting ( is large), put a lot weight on the new observation (K is large). If the new information is noisy (R is large), put less weight on the new information (K is small).

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Kalman Filter: Example Kalman gain is Consider State equation Observation equation Additionally , where is a constant Assume that we picked (we don’t know anything about ). Can you calculate the Kalman gain in the 1st period, ? What is the interpretation?

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Kalman Filter: The last step How do we get from to using ? Recall that for a bivariate normal distribution Using this property and the fact that Thus, st|t = st|t-1+bPt|t-1(Ft|t-1)-1(yt - yt|t-1) and Pt|t = Pt|t-1 – bPt|t-1(Ft|t-1)-1bPt|t-1

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Kalman Filter: Finally From the previous slide st|t = st|t-1+bPt|t-1(Ft|t-1)-1(yt - yt|t-1) Pt|t = Pt|t-1 – bPt|t-1(Ft|t-1)-1bPt|t-1 Need: from to using Thus, we get the expression for the Kalman gain: Similarly And we are done!

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Kalman Filter: Review We start from and . yt|t-1 = axt + bst|t-1 Calculate Kalman gain Update using observed Construct forecasts for the next period Repeat!

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Kalman Filter: How to choose initial state If the sample size is large, the choice of the initial state is not very important In short samples can have significant effect For stationary models Where Solution to the last equation is Why? Under some very general conditions as

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Kalman Filter as a Recursive Regression Consider a regular regression function where Substituting From one of the previous slides: st|t = st|t-1+bPt|t-1(Ft|t-1)-1(yt - yt|t-1)

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Kalman Filter as a Recursive Regression Thus the Kalman filter can be interpreted as a recursive regression of a type where is the forecasting error at time t The Kalman filter describes how to recursively estimate

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Optimality of the Kalman Filter Using the property of OLS estimates that constructed residuals are uncorrelated with regressors for all t Using the expression for and the state equation, it is easy to show that for all t and k=0..t-1 Thus the errors do not have any (linear) predictable component!

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Kalman Filter Some comments Within the class of linear (in observables) predictors the Kalman filter algorithm minimizes the mean squared prediction error (i.e., predictions of the state variables based on the Kalman filter are best linear unbiased): If the model disturbances are normally distributed, predictions based on the Kalman filter are optimal (its MSE is minimal) among all predictors: In this sense, the Kalman filter delivers optimal predictions.