r−+−−−+−−−−+++=−++−+−=−=111211......()()()()()()()()0,covvar,cov...var1,covvarvar11==−==−−+−mmtmttmmtmttmtmttmtrrrrrrmrrrmrThe autocorrelations decay to zero after m-1 lags.This is indicative of an MA(m-1) process.... ... ...1111111++=+++=+++=−−+−+−−+−−−+−mtmtmmtmtmttmtmtttmtrrrrrrrrrrrKlog pricesWarwick Business School 8
Method of Moments Estimation (MME)
OLS as a MMEThe CLRM requires the following populationmoment conditionsThe MME finds by solving the sample moment conditionsThere are k sample moment conditions and kunknown parameters ⇒possible to find a unique solution for . εβ+=Xy()0=′εXE()0ˆ1ˆ1=−′=′βεXyXTXTβˆ
This is a k×1 vector.⇒There are k moment conditions which the OLS estimator must satisfy.These moment conditions imply the values of Xare determined outside of the model:X is exogenousβˆ
()()error term theoft independenlinearly is X00 :A2=′⇒=εεXEXEWarwick Business School 9
OLS as a MME()()yXXXXXyXXyXT′′=⇒′=′⇒=−′−1ˆˆ0ˆ1βββThe MM estimator for the CLRM is identicalto the OLS estimator.Warwick Business School 10
Properties of MME
MME is a general approach to estimation which imposes population moment conditions (required by the statistical model) to hold exactly in the sample.
These moment conditions are then solved for the unknown parameters in the model (example above).
MME has 3 attractive features:
1.It makes no distributional assumptions.
2.It is a consistent estimator.
3.It is a verygeneral technique (e.g., applicable to non-linear models).Warwick Business School 11
Endogenous regressors
In many instances in economics/finance there is a two way or simultaneous relationship between X and y.⇒both X and y are determined insidethe model.⇒X is endogenous.Endogeneityis common due to the non-experimentalnature of economic/finance data ⇒In that case OLS/MME estimation (assuming ) is invalid. The estimator is biased (and inconsistent)⇒()0=′εXE()()()()εβεββXXXXXXXyXXX′′+=+′′=′′=−−−111ˆ()()()().0 unless ˆ1=′≠′′+=−εβεββXEXEXXE()0≠′εXEWarwick Business School 12
Examples of endogeneityin finance: testing CIP and UIP (Cuthberston& NitzscheChps24.3/4, 25.1/2)
Covered Interest Parity
Where Fh(h-period forward exchange rate), S(spot exchange rate) are denominated in terms of the domestic currency price of a unit of foreign exchange.
r(domestic interest rate), r*(foreign interest rate) (interest rate on h-period T-Bills).*11ttthtrrSF++=⇒Warwick Business School 13
CIP implies you can’tearn abnormal profits from…
1.Borrowing £x at the rate r.2.Converting £$ at the spot rate S:3.Investing in h-period US bonds at the rate r*:4.Simultaneously switching $’s back to £’s at the h-period forward rate Fh:5.Given risklessarbitrage (the investors receipts are covered in the forward market) would expect under EMH that:()*1$rSx+Sxx$£→()*1£rxSFh+()()*11r*11rSFrxrxSFhh+++=+=⇒Warwick Business School 14
Uncovered Interest Parity
Similar idea to CIP but with the keydifference that investors are willing to take a bet on what the exchange rate will be at the time of converting $’s back to £.
The investment in $’s is riskybecause the £receipts
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