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imperfect observation: there is a lack of natural units for ‘happiness’ or ‘utility’ whichrenders the scale of β inherently ambiguous. In some cases, where there are naturalunits of measurement for y * , we can fix σ ε at a reasonable hypothetical value. Forexample, for an analysis of survey responses to a question about expected inflation wemight reasonably set σ ε at (say) half a percentage point to allow a rough but directinterpretation of the model in terms of the natural units. Note that α is identifiableindependently of σ ε . As a consequence, we can estimate unambiguously the speed ofadjustment. For example, following a shock, the proportion of disequilibrium which iseliminated within s periods is 1-α s and this is unaffected by normalisation.2.3 DynamicsThe SD and LAR processes (1) and (2) imply different patterns of dynamic behaviour.Consider the following artificial example:SD model:y*t= .8yt−10 + x + ε(8)tLAR model:y*t= 0.422y−+ 0.355 + 0. 770x+ ε*t 1t(9)where x = 0.5, ε t ~ N(0,1) and y 1 ( y* > 0). The parameters of the LAR process (9)t=thave been chosen to reproduce exactly three properties of the SD process (8): 2(i) Pr(y = 1) = 0.877;(ii) ∂Pr(y = 1 | x)/∂x = 0.246;(iii) Pr(y t ≠ y t-1 ) = 0.170.With the LAR parameters chosen in this way, the distributions of run lengths in states0 and 1 are identical for the two processes. However, the relationship betweensuccessive run lengths is not. This is reflected in the autocorrelation functions (Figure1). As we would expect, the LAR model has much higher autocorrelations than theSD model for*yt. For the observed y t , the ACF decays faster for the SD than the LARprocess, despite the fact that they have the same 1st-order autocorrelation byconstruction. Thus, an LAR model will display greater persistence than anobservationally similar SD model, in this quite subtle sense.2 Conditions (i) and (ii) are imposed analytically to determine β 0 and β 1 for given α; Monte Carlosimulation was then used to find the value of α to equalise the 1 st order autocorrelations.- 4 -
0.450.40.350.30.250.2SD model: ACF(y)SD model: ACF(y*)LAR model: ACF(y)LAR model: ACF(y*)0.150.10.0501 2 3 4 5 6 7LAG ORDERFigure 1ACFs for the SD and LAR modelsThe two models also differ in terms of the implied dynamic multiplier effectsof x on y. To illustrate this, consider again the binary case and focus on two importantfeatures: the impact on Pr(y it =1 | y it-1 , X i , u i ) of switching the conditioning event fromy it-1 = 0 to y it-1 = 1; and the impact of the history of {x it } on the probability of apositive response, without conditioning on y it-1 .For the former, the SD model is relatively simple:Pr( yit= 1| yit−1= 1, Xi, ui) − Pr( yit= 1| yit−1= 0, Xi, ui) =Φ( α + β'x + u ) − Φ( β'x + u )where Φ(.) is the cdf of the N(0,1) distribution. For the LAR model, we have instead:itiiti(10)Pr( yit= 1| yit−1= 1, X , u ) − Pr( yiiit= 1| yit−1= 0, Xi, u )i=Pr( yitPr( y= 1, yit−1it−1= 1| X , u )= 1| X , u )iiii−Pr( yit= 1| X , u ) − Pr( yi1−Pr( yiit−1it= 1, yit−1= 1| X , u )ii= 1| Xi, u )i=Pr( yit= 1, yit−1Pr( y= 1| X , u ) − Pr( yit−1ii= 1| X , u ) 1iiit= 1| X , u ) Pr( yi[ − Pr( y = 1| X , u )]it−1iiit−1i= 1| X , u )ii(11)Assume the process (2) is stable and long-established. Then:y*it=∞∑s=0∞s uisα β ' xit− s+ + ∑αεit−s(12)1−αs=0- 5 -
Page 1 and 2: ESTIMATION OF DYNAMIC LINEAR MODELS
Page 3 and 4: 1 IntroductionIn discrete data mode
Page 8 and 9: and therefore Pr(y it =1, y it-1 =1
Page 10 and 11: y*itt= α y*i0+t−1∑s=0s 1−αs
Page 18: then done sequentially, starting wi
Page 21 and 22: 7 ConclusionsWe have considered an
Page 23 and 24: *Appendix 1: A simulation approxima
Page 25: Appendix 2: SD model estimatesTable

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