Figure 2.3: Inflation Rates of Germany, France, Italy and the UK
Figure 2.4: Unemployment Rates of Germany, France, Italy and the UK
behaviors have been considered. The evidence of the countries studied shows that there do exist some significant relations between the output gap and real interest rate, and between the inflation and the output gap. In order to explore regime changes in the economy I have also estimated a time-varying Phillips curve. The estimation results show that the reaction to the unem-ployment gap has been changing, indicating regime changes in the economy.
Appendix A: The State-Space Model and Kalman Filter
Here I make a brief sketch of the State-Space model (SSM) and Kalman filter, following Harvey (1989, 1990) and Hamilton (1994).7 After arranging a model in a State-Space form, one can use the Kalman filter to obtain the paths of time-varying parameters.
The State-Space Model
The State-Space model applies to a multivariate time series, yt, containing N elements. These observable variables are, via a so-called “measurement equation”, related to anm×1 vector,αtwhich is known as the “state vector”, yt=Ztαt+dt+ǫt, (2.30) with t = 1, ..., T, Zt is an N ×m matrix, dt is an N ×1 vector and ǫt is an N ×1 vector of serially uncorrelated disturbances with zero mean and covariance matrix Ht. Usually the elements of αt are not observable but are known or assumed to be generated by a first-order Markov process, which is known as the “transition equation”
αt=Ttαt−1+ct+Rtηt, (2.31) with t= 1, ...T. Ttis an m×m matrix, ct is anm×1 vector,Rtis an m×g matrix andηtis ag×1 vector of serially uncorrelated disturbances with zero mean and covariance Qt. If the system matrices Zt, dt, Ht, Tt, ct, Rt and Qt
do not change over time, the model is said to be time-invariant, otherwise, it is time-variant.
The Kalman Filter
The Kalman filter estimates time-varying parameters in three steps. Given all the information currently available, the first step forms the optimal predictor
7Although there are numerous books dealing with the Kalman filter, the framework in this appendix is mainly based on Harvey (1989, 1990).
of the next observation via the so-called “prediction equations”. The second step is to update the estimator by incorporating the new observation via the “updating equations”. These two steps use only the past and current information, disregarding the future information which may also affect the estimation. Therefore, the third step is to “smooth” the estimators based on all of the observations to get a more reasonable result.
Prediction Let at−1 denote the optimal estimate of αt−1 based on the observations up to and includingyt−1. LetPt−1 denote them×mcovariance matrix of the estimate error, i.e.
Pt−1 =E[(αt−1−at−1)(αt−1−at−1)′].
Given at−1 and Pt−1, the optimal estimate of αt is given by
at|t−1 =Ttat−1+ct, (2.32)
while the covariance matrix of the measurement error is
Pt|t−1 =TtPt−1Tt′+RtQtRt′, t= 1, ..., T. (2.33) These two equations are called the prediction equations.
Updating Once the new observations ofyt become available, the estimate of αt, at|t−1, can be updated with the following equations
at=at|t−1+Pt|t−1Zt′Ft−1vt, (2.34) and
Pt =Pt|t−1−Pt|t−1Zt′Ft−1ZtPt|t−1, (2.35) where vt =yt−Ztat|t−1−dt, which is called the prediction error, and Ft = ZtPt|t−1Zt′+Ht, fort = 1, ..., T.
Smoothing The prediction and updating equations estimate the state vec-tor, αt, conditional on the information available at time t. The aim of
smoothing is to take account of the information available after time t.8 The smoothing algorithms consist of a set of recursions that start with the final quantities (aT and PT) and work backwards. The equations are
at|T =at+Pt∗(at+1|T −Tt+1at−ct+1), (2.36) and
Pt|T =pt+Pt∗(Pt+1|T −Pt+1|t)Pt∗′, (2.37) where
Pt∗ =PtTt+1′ Pt+1|t−1 , t=T −1, ...,1, with aT|T =aT and PT|T =PT.
The Maximum Likelihood Function In order to estimate the state vec-tor, one must first estimate a set of unknown parameters (n×1 vector ψ, referred to as “hyperparameters”) with the maximum likelihood function.
For a multivariate model the maximum likelihood function reads L(y;ψ) =
T
Y
t=1
p(yt|Yt−1),
wherep(yt|Yt−1) denotes the distribution ofytconditional on the information set at time t−1, that is, Yt−1 = (yt−1, yt−2, ..., y1). The likelihood function for a Gaussian model can be written as
logL(ψ) =−(1/2)(N T log2π+
T
X
t=1
log|Ft|+
T
X
t=1
vt′Ft−1vt), (2.38) where Ft and vt are the same as those defined in the Kalman filter.
In sum, one has to do the following to estimate the state vector with the Kalman filter. (a) Write the model in a State-Space form of Eq. (2.30)-(2.31), run the Kalman filter of Eq. (2.32)-(2.35) and store all vt and Ft for future use. (b) Estimate the hyperparameters with the maximum likelihood
8Harvey (1989) points out three smoothing algorithms: “Fixed-point” smoothing,
“Fixed-lag” smoothing and “Fixed-interval” smoothing. In this dissertation I use the third one, which is widely used in economic problems.
function presented in Eq. (2.38). (c) Run the Kalman filter again with the estimates of the hyperparameters to get the non-smoothed estimates of the state vector. (d) Smooth the state vector with the smoothing equations Eq.
(2.36)-(2.37).
In order to run the Kalman filter one needs starting values of at and Pt, that is, one needs to know a0 and P0. For a stationary and time invariant transition equation, the starting values are given as follows:
a0 = (I−T)−1c, (2.39)
and
vec(P0) = [I−T ⊗T]−1vec(RQR′). (2.40) If the transition equation is non-stationary, the initial conditions must be estimated from the model. There are usually two approaches to deal with this problem. The first approach assumes that the initial state is fixed withP0 = 0 (or a zero matrix) and the initial state is treated as unknown parameters that will be estimated from the model. The second approach assumes that the initial state is random and has a diffuse distribution, that is, its covariance matrix is P0 =κI, with κ being a large number.
Time-Varying Coefficient Estimation Consider a linear model yt=x′tβt+ǫt, t= 1, ..., T,
where xtis a k×1 vector of exogenous variables andβt is the corresponding k × 1 vector of unknown parameters which evolve over time according to certain stochastic processes. Defining βt as the state vector, one can use the State-Space model and Kalman filter to estimate the time-varying param-eters. There are basically three classes of models that can be used for the time-varying coefficient estimation:
The Random-Coefficient Model In this model the coefficients vary ran-domly about a fixed, but unknown mean, ¯β. The State-Space form is
yt =x′tβt
βt= ¯β+ǫt, ǫt ∼N ID(0, Q),
for all t. The time-varying coefficients in this model are stationary and do not show structural changes.
The Random-Walk Model In the random-walk model the coefficients are non-stationary and follow a random-walk path. The State-Space form reads:
yt=x′tβt+ǫt, t= 1, ..., T
where ǫt∼N ID(0, H) and the vector βt is generated by the process βt =βt−1+ηt,
where ηt∼N ID(0, Q).
The Return-to-Normality Model In this model the coefficients are gen-erated by a stationary multivariate AR(1) process. The State-Space form reads
yt=x′tβt+ǫt, t= 1, ..., T, (2.41) βt−β¯=φ(βt−1−β) +¯ ηt, (2.42) where ǫt∼N ID(0, H), andηt ∼N ID(0, Q). The coefficients are stationary and evolve around a mean, ¯β. It is clear that the random-coefficient and random-walk models are just two special cases of the return-to-normality model.
In order to apply the Kalman filter one has to rearrange the return-to-normality model in a standard State-Space form. Let βt∗ =βt−β, one has¯
yt= (x′t x′t)αt+ǫt, t= 1, ..., T (2.43) and
αt=
"
β¯t
βt∗
#
=
"
I 0 0 φ
# "
β¯t−1
βt−1∗
# +
"
0 ηt
#
. (2.44)
A diffuse prior is used for ¯βt, implying that the starting values are constructed from the firstkobservations. The starting value ofβt∗is given by a zero vector with the starting covariance matrix given by Eq. (2.40).
Appendix B: Derivation of the New Keynesian Phillips Curve by Woodford (1996)
Here I make a brief sketch of Woodford’s (1996) derivation of the New Key-nesian Phillips (and IS) curve. The details of the derivation can be found in Woodford (1996, p.3-14).
The economy consists of a continuum of identical infinite-lived house-holds indexed by j ∈[0,1], and z ∈[0,1] denotes a continuum of differenti-ated goods produced by the households. The objective of each household is assumed to maximize the following function
E ( ∞
X
t=0
βthu(Ctj+Gt) +v(MtJ/Pt)−ω[yt(j)]i )
, (2.45)
whereuandv are increasing concave functions and ωis an increasing convex function. β denotes the discount factor between 0 and 1. yt(j) denotes the product supplied by household j. The term v “indicates the existence of liquidity services from wealth held in the form of money” (Woodford, 1996, p.5). Ctj is the consumption of householdj
Ctj ≡ Z 1
0
cjt(z)θ−1θ dz θ−1θ
, (2.46)
wherecjt(z) denotes householdj’s consumption of goodzat timet, andθ >1 is the constant elasticity of substitution among alternative goods. Gtdenotes the public goods. Mtj denotes the household’s money balances at the end of period t, and Pt is the price index of goods
Pt≡ Z 1
0
pt(z)1−θdz 1−θ1
, (2.47)
with pt(z) being the price of goodz at timet. The budget constraint of each household reads
Z 1 0
pt(z)cjt(z)dz+Mtj+Et(Rt,t+1Bt+1j )≤Wtj+pt(j)yt(j)−Tt, (2.48)
where Bjt+1 denotes the bond portfolio at date t and Rt,T is the stochastic discount factor. Wtj denotes the nominal value of the household’s financial wealth at the beginning of period t, that is,
Wtj =Mt−1j +Btj, (2.49) andTtis the net nominal lump-sum tax. Woodford (1996, p.6) further claims that the budget constraint (2.48) is equivalent to the following expression
∞
X
T=t
Et
Rt,T[
Z 1 0
pT(z)cjT(z)dz+ iT
1 +iT
MTj
≤
∞
X
T=t
Et{Rt,T[pT(j)yT(j)dz−TT]}+Wtj, (2.50) with it denoting the nominal interest rate on a riskless bond, therefore
1 +it≡ 1
Et(Rt,t+1). (2.51)
The consumption of good z in line with expenditure minimization and the demand of good j in line with cost minimization turn out to be
cjt(z) =Ctj
pt(z) Pt
−θ
(2.52) and
yt(j) = Yt
pt(j) Pt
−θ
, (2.53)
with Yt = Ct+Gt and Ct = R1
0 Cthdh. Woodford (1996, p.7) further gives three necessary and sufficient conditions for an optimal consumption and portfolio plan of a household, that is,
βT−tu′(YT) u′(Yt)
Pt
PT
=Rt,T (2.54)
v′(Mt/Pt) u′(Yt) = it
1 +it
(2.55) and that (2.50) holds with equality at date 0. From (2.54) one knows
βEt
u′(Yt+1) u′(Yt)
Pt
Pt+1
= 1
1 +it
. (2.56)
Next, Woodford (1996, p.8-9) shows how to set the price. Following the Calvo (1983) price-setting model, namely, each period a fraction 1 −α of goods suppliers set a new price and the remaining α keep the old price, Woodford (1996, p.8) shows that the price p must be set to maximize
∞
X
k=0
αk{ΛtEt[Rt,t+kpyt+k(p)]−βkEt[ω(yt+k(p))]},
with yT(p) being the demand at date T given by (2.53). Λt denotes the marginal utility of holding money. The optimal price Pt satisfies the first-order condition
∞
X
k=0
αkEt{Rt,t+kYt+k(Pt/Pt+k)−θ[Pt−µSt+k,t]}= 0, (2.57) where µ≡ θ−1θ and ST,t denotes the marginal cost of production at dateT:
ST,t= ω′[YT(Pt/PT)−θ]
u′(YT) PT. (2.58)
Employing Eq. (2.47), one finds that
Pt = [αPt−11−θ+ (1−α)Pt1−θ]1−1θ. (2.59) On the basis of the analysis above, Woodford (1996) then explores how fiscal policy may affect macroeconomic instability. I will not sketch his analysis of this problem here, since this is not of much interest in my dissertation.
Defining xt as the percentage deviation of Yt from its stationary value Y∗ (namely, xt = YtY−Y∗∗) and ˆπt as the percentage deviation of πt from its sta-tionary value,9 and linearizing (2.56) at the stationary values of Yt, πt and it, one then obtains the following IS curve10
xt =Etxt+1−σ(ˆit−Etπˆt+1), (2.60) with ˆit being the percentage deviation of the nominal interest rate from its stationary value, and
σ ≡ − u′(Y∗) u′′(Y∗)Y∗.
9πtis defined as PPtt
−1, since the stationary value ofπtis 1, ˆπtis then equal to PtP−tPt−1
−1 .
10The stationary value of itis found to beβ−1−1.
After linearizing Eq. (2.57)-(2.59) around the stationary values of the vari-ables and rearranging the terms, one obtains
Pˆt= κα 1−α
∞
X
k=0
(αβ)kEtxt+k+
∞
X
k=1
(αβ)kEtπˆt+k, (2.61) ˆ
πt=1−α
α Pˆt, (2.62)
with
κ≡ (1−α)(1−αβ) α
̟+σ
σ(̟+θ) and ̟≡ ω′(Y∗) ω′′(Y∗)Y∗,
where ˆPtis the percentage deviation ofPt/Ptfrom its stationary value, which is 1. After rearranging Eq. (2.61) as
Pˆt=αβEtPˆt+1+ κα
1−αxt+αβEtπˆt+1 (2.63) and substituting (2.62) into (2.63), one finally obtains the following Phillips curve:
ˆ
πt=βEtπˆt+1+κxt. (2.64)