Chapter 3 is organized as follows. Section 2 addresses the framework and methodology of the data rich approach using the DSGE model. Section 3 describes the SW model. Section 4 explains Bayesian estimation methods in a data-rich approach. Section 5 explains the preliminary settings and data description. Section 6 presents the estimation results. Section 7 concludes.
3.2 Data Rich Approach
DFMs are rapidly developed and applied in many fields of macroeconomics. One of applications is to estimate the DSGE model in a data rich environment. The main idea of data rich estimation is (1) to extract the common factor from a large panel of macroeconomic time series data, then (2) to match the state variable with the extracted common factor. This section illustrates the so-called data rich estimation method proposed by Boivin and Giannoni (2006) and Kryshko (2011), and describes how to utilize a large number of data in estimating DSGE model.
DFM is represented by the state space model consisted of three linear equations: The equation (3.8) is measurement equation, the equation (3.9) is state equation and (3.10) is the transition equations of measurement error. LetFtdenote theN×1 vector of the unobserved common factor, and Xt denote the J ×1 vector of the large panel of data. Note that the number of data series shall be much larger than the number of state variables in a data rich environment, i.e. J N.
Xt
|{z}
J×1
= |{z}Λ
J×N
Ft
|{z}
N×1
+|{z}et J×1
, (3.8)
Ft
|{z}
N×1
= |{z}G
N×N
Ft−1
| {z }
N×1
+ εt,
|{z}
N×1
εt∼i.i.d.N(0,Q), (3.9)
et
|{z}
J×1
= |{z}Ψ
J×J
et−1
|{z}
J×1
+|{z}νt J×1
νt∼i.i.d.N(0,R), (3.10)
whereΛis the factor loadings matrix (J×N),etis the idiosyncratic errors vector which is allowed to be serially correlated as shown in the equation (3.10). G is N ×N matrix, and the common factor Ft follows the AR(1) process (3.9). The matrices Ψ, Q and R are assumed to be diagonal in the exact DFM as in Stock and Watson (2005).
DSGE models are also expressed by the state space models as the following three equations:
Xt
|{z}
J×1
= |{z}Λ
J×N
S¯t
|{z}
N×1
+ et
|{z}
J×1
, (3.11)
S¯t
|{z}
N×1
= G(θ)
| {z }
N×N
S¯t−1
| {z }
N×1
+H(θ)
| {z }
N×M
εt,
|{z}
M×1
εt∼i.i.d.N(0,Q(θ)), (3.12)
et
|{z}
J×1
= Ψt
|{z}
J×J
et−1
|{z}
J×1
+ νt
|{z}
J×1
νt∼i.i.d.N(0,R), (3.13)
whereXtis the observable variables vector (J×1), ¯Stis the state variables vector (N×1), andεt is the structural shocks vector (M ×1).
DSGE models can be estimated using the Kalman filter, as can DFMs. Thus, we can apply the estimation method for the DFMs to DSGE models. However, the major difference between DFMs and a DSGE models is meanings of parameters. The DSGE model has microeconomic foundations in which agents in the model solve the dynamic optimization problem, given the structural parameters.
The structural parameters can be interpreted as tastes of households or technologies of firms that are not affected by the fiscal and monetary policies. In contrast, DFM’s (reduced-form) parameters are not necessarily interpretable. The solution of the model (the law of motion around steady states) can be expressed by (3.12). We will refer to the elements of ¯St = [x0t,S0t]0 as “model concepts,”
e.g. output gap, inflation gap and so on. zt is a vector of non-predetermined endogenous variables (or jump variables) and St is a vector of predetermined endogenous and exogenous variables. εtis a vector of exogenous shocks. θ is the structural parameters and G(θ) is a matrix of parameters.
Using the DFM framework, the data rich DSGE model can be represented as (3.11), (3.12) and (3.13). In contrast with the regular DSGE model, there is a many-to-many relation between Xt
and S¯t, since the matrix Λ is J ×N (J N) in (3.11), which could grasp the theoretical gap between the data indicator Xtand the model concept ¯St.
In a regular DSGE model, the number of observable variable is lesser than (or equal) that of state variables, i.e.J ≤ N. In the data rich DSGE model, the number of observable variables is much larger than that of state variables (J N) as well as a DFM. From (3.11) and (3.12), the components ΛS¯t will be consistent with microeconomic theory. By contrast, the measurement error et indicates a “specific (or unique)” factor of the corresponding data Xt. In other words, the measurement error does not follow economic theory nor is it affected by fluctuations of other data and other endogenous variables (model concepts). Because it is allowed to fluctuate freely, the measurement errors might play an important role in removing undesirable relations between the observable variables and the model concept variables caused by model misspecification or the mismatch between model concepts and appropriate observable variables.
In addition, there is also technical requirement to introduce measurement errors. Without measurement errors, the stochastic singularities should be inevitable. Usually, as long as the number of data series does not exceed the number of structural shocks, the DSGE model can be estimated without measurement errors. But if it does not hold, we cannot estimate the model: According to (3.12), state variables (model concepts) will be represented by the linear combination of structural shocksεt. Then, the state variable is connected into the data series through (3.11), so data should also be expressed by the linear combination of the structural shocks. Normally, all state variables are not considered observable, thus the number of state variables is larger than the number of data.
However, if the number of structural shocks is less than the data series, thus, if the number of structural shocks is less than the number of the state variables, the state variables will be linearly dependent with each other. As a consequence, the variance covariance matrix of state variables becomes singular, so we cannot evaluate the likelihood (stochastic singularity). Especially in a data rich environment, if we do not deal with anything, a large number of data series will have to exceed the number of structural shocks greatly, and we cannot estimate the model due to the problem.
Therefore, we need to introduce measurement errors when utilizing a large data set to estimate the DSGE model.
One of advantages of data rich approach is to identify structural shocks and measurement errors through state space model. It might be hard to identify between them in one-to-one matching of the data and the model concepts, as usual estimation procesure of the DSGE model. However, by connecting the model concept with the common factor extracting from many data, the unexplainable factor by the model (i.e. measurement error) can be easily separated. One of our aims is to verify whether this advantage of data rich estimation methods work successfully.
The structural shocks εt and disturbance terms νt in measurement errors et follow normal distributions,εt∼i.i.d.N(0,Q(θ)) andνt∼i.i.d.N(0,R), respectively. Their variance covariance matricesQ(θ) andRare positive definite and diagonal matrices. The AR(1) coefficientsΨin (3.13)
3.2. DATA RICH APPROACH 87 are also assumed to be a diagonal matrix. These assumptions imply that measurement errorset are independent of each other in terms of cross section but are dependent on their own lag variables in terms of time series. Finally, it should be noted that, in the DSGE model, the matricesG(θ),H(θ) and Q(θ) are derived as the solution of the model, and become functions of structural parameters θ.
In the DFM framework, state variables Ftare regarded as common factors from the viewpoint of statistics, as shown by (3.8). Hence, there is no need to care about how Ft and Xt are tied, that is, we do not care about the magnitudes and signs of elements of the factor loadings matrixΛ.
In the data rich DSGE framework, data indicatorsXt are matched with model concepts ¯St in the measurement equation (3.8) and (3.11). In this case, we have to mind (a part of) the specification of the coefficient matrixΛ in (3.11). Otherwise, we cannot identify the model concepts ¯S1,t:
Let us denote a model concept (for example, output gap) as ¯S1,tand suppose that ¯S1,tis matched with two data indicators [X1,t1 , X2,t1 ]0 (for instance, GDP data and IIP data). Then, we can express the measurement equation as follows:
X1,t1 X2,t1
= λ11
λ12
S¯1,t+ e11,t
e12,t
where λ11 and λ12 are factor loadings coefficients, and e11,t and e12,t are measurement errors. Thus, X1,t1 (GDP data) and X2,t1 (IIP data) have a common factor ¯S1,t (output gap). Now, let us regard the equation above as a regression model in which the two data are regressed by a common factor S¯1,t. In this case, the two data are linearly dependent with each other, so that the multi-collinearity problem arises. As is well known, the problem makes the estimated coefficients, λ11 and λ12, to be unstable. Moreover, we need to estimate not only factor loadings parameters but also state variable ( ¯S1,t) itself. The unstable factor loadings coefficient also causes fragile estimates of state variables.
To avoid the problem above, we impose additional restrictions on factor loadingsΛ. In the above example, we set λ11 = 1 andλ12 to be estimated. In other words, output gap ( ¯S1,t) is fundamentally explained by the GDP data (X1,t), but if the IIP data (X2,t) has some additional information on the output gap, the marginal increment can be estimated as λ2,1.
X1,t1 X2,t1
= 1
λ12
S¯1,t+ e11,t
e12,t
Next, consider the case where there are two model concepts (for example, output gap and inflation gap). Suppose inflation gap ¯S2,t is fundamentally explainable by GDP deflatorX1,t2 , but CPI data X2,t2 is also assumed to have some information on inflation gap:
X1,t1 X2,t1 X1,t2 X2,t2
=
1 0
λ12 0
0 1
0 λ22
S1,t
S¯2,t
+
e11,t e12,t e21,t e22,t
Furthermore, we have an extra data Xtinf o might affect both model concepts (output gap and inflation gap). For example, stock price data could be regarded as the extra data. Then, the measurement equation can be expressed by:
X1,t1 X2,t1 X1,t2 X2,t2 Xtinf o
=
1 0
λ12 0
0 1
0 λ22 λ1inf o λ2inf o
S1,t
S¯2,t
+
e11,t e12,t e21,t e22,t einf ot
Again, let us regard the equation above as a regression model. The final equation is Xtinf o = λ1inf oS¯1,t+λ2inf oS¯2,t+einf ot , which implies the coefficientλ1inf o can be interpreted as the marginal contribution for ¯S1,t (output gap) of explaining Xtinf o (stock price data), by controlling the effect of the ¯S2,t (inflation gap). Conversely, the information on the additional data Xtinf o is reflected to estimate the output gap ¯S1,t. In this way, utilizing the extra data could contribute to accurate estimation of model concepts.
Finally, suppose that we have an another model concept ¯S3,t (for example, shadow price of capital) and the information seriesX1,tinf o (stock price) also has some beneficial information for the shadow price. Also,Xtinf owill affect all model concepts (output gap, inflation gap and shadow price of capital). Then, with some rearrangement, we can write the measurement equation as follows:
X1,t1 X1,t2 X2,t1 X2,t2 Xtinf o
=
1 0 0
0 1 0
λ12 0 0
0 λ22 0
λ1inf o λ2inf o λ3inf o
S1,t
S¯2,t S¯3,t
+
e11,t e12,t e21,t e22,t einf ot
whereλ3inf o is the factor loadings coefficient on ¯S3,t.
According to this idea, we categorize the data indicator Xt into two types. The first type is referred to as sensor series which is likely to embody a certain model concept and link to just one variable of the concept. In the above example, X1,t1 (GDP data), X2,t1 (IIP data), X1,t2 (GDP deflator) andX2,t2 (CPI) correspond to sensor series indicators. The sensor series is the data modeled as to how it relates to the model concepts. The other type is referred to asinformation series. This data is not directly connected to any specific model concept but seems to hold useful information on a number of model concepts. In this example, it is Xtinf o (stock price data). The classification of data indicator depends on how the factor loadings matrixΛ is specified.
Also, unlike the common factor Ft in DFM, the specification of factor loadings matrix leads to the classification of the model concepts ¯St into two types: One can be considered to directly correspond to data indicators ( ¯Slinkt ). In the example above, ¯S1,t (output gap) and ¯S2,t (inflation gap). The other does not directly link to data indicator ( ¯Snont ). For example, model concepts such as shadow price of physical capital ( ¯S3,t) or rental rate of phisical capital are not necessarily clear as to which data indicator to match. In sum, to capture types of data indicators and model concepts, we rewrite the measurement equation (3.11) as (3.14):
Xsensort
− − − − − − −−
Xinf ot
| {z }
J×1
=
Λsensor
− − − − −−
Λinfo1
| O
−− − − −−
| Λinfo2
| {z }
J×N
S¯linkt S¯nont
| {z }
N×1
+ et
|{z}
J×1
, (3.14)
where Xsensort and Xinf ot indicate sensor series and information series, respectively. The ma-trices Λsensor, Λinf o1 and Λinf o2 are partitioned matrices of the factor loading matrix Λ. Let S¯linkt and ¯Snont can be represented by Nlink ×1 and Nnon ×1 vectors of model concepts, re-spectively. And suppose that we will link each model concept in ¯Slinkt with p sensor series data, Xsensort = [X1,tsensor, X2,tsensor,· · · , Xp,tsensor]0. Then, we can express the factor loadings matrixΛsensor as follows: