# Stability. Coefficients may change over time. Evolution of the economy Policy changes

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1 Sabiliy Coefficiens may change over ime Evoluion of he economy Policy changes

2 Time Varying Parameers y = α + x β + Coefficiens depend on he ime period If he coefficiens vary randomly and are unpredicable, hen hey canno be esimaed As here would be only one observaion for each se of coefficiens We canno esimae coefficiens from jus one observaion! e

3 Smoohly Time Varying Parameers y = α + x β + If he coefficiens change gradually over ime, hen he coefficiens are similar in adjacen ime periods. We could ry o esimae he coefficiens for ime period by esimaing he regression using observaions [ w/2,, + w/2] where w is called he window widh. w is he number of observaions used for local esimaion e

4 Rolling Esimaion This is called rolling esimaion For a given window widh w, you roll hrough he sample, using w observaions for esimaion. You advance one observaion a a ime and repea Then you can plo he esimaed coefficiens agains ime

5 Wha o expec Rolling esimaes will be a combinaion of rue coefficiens and sampling error The sampling error can be large Flucuaions in he esimaes can be jus error If he rue coefficiens are rending Expec he esimaed coefficiens o display rend plus noise If he rue coefficiens are consan Expec he esimaed coefficiens o display random flucuaion and noise

6 Example: GDP Growh

7 STATA rolling command STATA has a command for rolling esimaion:.rolling, window(100) clear: regress gdp L(1/3).gdp In his command: window(100) ses he window widh w=100 The number of observaions for esimaion will be 100 clear Clears ou he daa in memory The daa will be replaced by he rolling esimaes I is necessary

8 rolling command.rolling, window(100) clear: regress gdp L(1/3).gdp The par afer he : regress gdp L(1/3).gdp This is he command ha STATA will implemen using he rolling mehod An AR(3) will be fi using 100 observaions, rolling hrough he sample

9 Example GDP, quarerly, 1947Q1 hrough 2009Q4 251 observaions Using w=100 The firs esimaion window is 1947Q2 1972Q1 The second is 1947Q3 1972Q2 There are 152 esimaion windows The final is 1985Q1 2009Q4

10 STATA Execuion:

11 Afer Rolling Execuion The original daa have been cleared from memory STATA shows new variables sar end _sa_1 _sa_2 _sa_3 _b_cons sar and end are saring/ending daes for each window sar runs from 1947Q2 o 1985Q1 end runs from o 1972Q1 2009Q4 The ohers are he rolling esimaes, AR and inercep

12 Time rese As he original daa have been cleared, so has your ime index. So he sline command does no work unil you rese he ime You can se he ime o be sar or end.sse sar.sse end Or, more eleganly, you can se he ime o be he midpoin of he window.gen =round((sar+end)/2).forma %q.sse This ime index runs from 1959Q4 hrough 1997Q3

13 Example Time rese example

14 Plo Rolling Coefficiens Now you can plo he esimaed coefficiens agains ime You can use separae or join plos.sline _b_cons.sline _sa_1 _sa_2 _sa_3

15 Rolling Inercep

16 Rolling AR coefficiens

17 Analysis The esimaed inercep is decreasing gradually The AR(1) coef is quie sable The AR(2) coef sars increasing around 1990 The AR(3) coef is 0 mos of he period, bu is negaive from and afer 1995 All of he graphs go a bi crazy over

18 Sequenial (Recursive) Esimaion As an alernaive o rolling esimaion, sequenial or recursive esimaion uses all he daa up o he window widh Firs window: [1,w] Second window: [1,w+1] Final window: [1,T] Wih sequenial esimaion, window is he lengh of he firs esimaion window

19 Recursive Esimaion STATA command is similar, bu adds recursive afer comma.rolling, recursive window(100) clear: regress gdp L(1/3).gdp STATA clears daa se, replaces wih sar, end, and recursive coefficien esimaes _b_cons, _sa_1, ec. Use end for ime variable.sse end This ses he ime index o he end period used for esimaion

20 Recursive Inercep

21 Recursive AR coefficiens

22 Analysis The recursive inercep flucuaes, bu decreases Drops around 1984, and 1990 The recursive AR(1) and AR(2) coefs are very sable The recursive AR(3) coef increases, and hen becomes sable afer 1984.

23 Summary Use rolling and recursive esimaion o invesigae sabiliy of esimaed coefficiens Look for paerns and evidence of change Try o idenify poenial breakdaes In GDP example, possible daes: 1970, 1984, 1990

24 Tesing for Breaks Did he coefficiens change a some breakdae *? We can es if he coefficiens before and afer * are he same, or if hey changed Simple o implemen as an F es using dummy variables Known as a Chow es

25 Gregory Chow Professor Gregory Chow of Princeon Universiy (emerius) Proposed he Chow Tes for srucural change in a famous paper in 1960

26 Dummy Variable For a given breakdae * Define a dummy variable d d=1 if >* Include d and ineracions d*x o es for changes

27 Model wih Breaks Original Model y = α + x β + e Model wih break y = α + x β + δd + γd x + Inerpreing he coefficiens δ=change in inercep γ=change in slope e

28 Chow Tes y = α + x β + δd + γd x + e The model has consan parameers if δ=γ=0 Hypohesis es: H 0 : δ=0 and γ=0 Implemen as an F es afer esimaion If prob>.05, you do no rejec he hypohesis of sable coefficiens

29 Example: GDP

30

31 Chow es The p value is larger han 0.05 I is no significan We do no rejec hypohesis of consan coefficiens

32 Fishing for a Breakdae An imporan rouble wih he Chow es is ha i assumes ha he breakdae is known before looking a he daa Bu we seleced he breakdae by examining rolling and recursive esimaes This means ha are oo likely o find misleading evidence of non consan coefficiens

33 Fishing We could consider picking muliple possible breakdaes *=[ 1, 2,, M ] For each breakdae *, we could esimae he regression and compue he Chow saisic F(*) Fishing for a breakdae is similar o searching for a big (significan) Chow saisic.

34 The Quand Likelihod Raio (QLR) Saisic (also called he sup Wald saisic) The QLR saisic = he maximal Chow saisics Le F(τ) = he Chow es saisic esing he hypohesis of no break a dae τ. The QLR es saisic is he maximum of all he Chow F- saisics, over a range of τ, τ 0 τ τ 1 : QLR = max[f(τ 0 ), F(τ 0 +1),, F(τ 1 1), F(τ 1 )] A convenional choice for τ 0 and τ 1 are he inner 70% of he sample (exclude he firs and las 15%. 34

35 Richard Quand Professor Richard Quand (1930 ) Princeon Universiy Esimaion of breakdae (Quand, 1958) QLR es (Quand, 1960)

36 QLR Criical Values QLR = max[f(τ 0 ), F(τ 0 +1),, F(τ 1 1), F(τ 1 )] Should you use he usual criical values? The large-sample null disribuion of F(τ) for a given (fixed, no esimaed) τ is F q, Bu if you ge o compue wo Chow ess and choose he bigges one, he criical value mus be larger han he criical value for a single Chow es. If you compue very many Chow es saisics for example, all daes in he cenral 70% of he sample he criical value mus be larger sill! 36

37 Ge his: in large samples, QLR has he disribuion, max a s 1 a q 2 1 Bi ( s) q i= 1 s(1 s), where {B i }, i =1,,n, are independen coninuous-ime Brownian Bridges on 0 s 1 (a Brownian Bridge is a Brownian moion deviaed from is mean), and where a =.15 (exclude firs and las 15% of he sample) Criical values are abulaed in SW Table

38 Noe ha hese criical values are larger han he F q, criical values for example, F 1, 5% criical value is

39 QLR Theory Disribuion heory for he QLR saisic Developed by Professor Donald Andrews (Yale)

40 Has he poswar U.S. Phillips Curve been sable? Consider a model of ΔInf given Unemp he empirical backwards-looking Phillips curve, esimaed over ( ): Δ Inf = ΔInf 1.37ΔInf ΔInf 3.04ΔInf 4 (.44) (.08) (.09) (.08) (.08) 2.64Unem Unem Unem Unemp 4 (.46) (.86) (.89) (.45) Has his model been sable over he full period ? 40

41 QLR ess of sabiliy of he Phillips curve. dependen variable: ΔInf regressors: inercep, ΔInf 1,, ΔInf 4, Unemp 1,, Unemp 4 es for consancy of inercep only (oher coefficiens are assumed consan): QLR = (q = 1). 10% criical value = 7.12 don rejec a 10% level es for consancy of inercep and coefficiens on Unemp,, Unemp 3 (coefficiens on ΔInf 1,, ΔInf 4 are consan): QLR = (q = 5) 1% criical value = 4.53 rejec a 1% level Break dae esimae: maximal F occurs in 1981:IV Conclude ha here is a break in he inflaion unemploymen relaion, wih esimaed dae of 1981:IV 41

42 42

43 Implemenaion I is difficul o compue QLR wihou using some programming. Bu i is well approximaed by Examining rolling and recursive esimaes for possible breaks Compuing Chow es a poenial breakdaes. Don use STATA s p value! Use Table 14.6 from SW (or earlier slide).

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