3.5.1 Measurement Equation
The observed variables should be regarded as imperfectly measured, i.e., they are contaminated by measurement errors. The measurement errors are unrelated to the model concepts. The usual methods without measurement errors might lead to biased parameters and structural shocks due to the endogeneity problem by omitting the measurement errors. One of the purposes of this research is to remove the measurement errors (unexplainable components by the model) from the data by extracting common factors from a large number of the data and matching the common factors with model concepts (explainable components by the model). That is, the data rich approach would bring unbiased parameters and structural shocks.
To this end, we consider the following four cases where different restrictions are imposed on the measurement equation (3.14), i.e., on the link between the model concepts ¯St and the data indicators Xt. Table 3.1 summarizes the four cases (Cases SW, A, B, and C).
(1) Case SW (Regular DSGE)
The first case is based on a regular estimation method of the DSGE model with a small data set (just seven indicators) in a fashion of one-to-one matching of data with model concept. This case just replicates the estimation result of SW (2003). The small data consists of real GDP, GDP deflator, nominal interest rate (call rate), investment, consumption, labor and real wage. The seven model concepts ¯Slinkt are assumed to be perfectly observed by data indicators. Hence, there is no measurement error in this model, i.e., et=0. The measurement equation(3.14) can be represented as follows:
Xsensor1,t = [I O]
S¯linkt S¯nont
(3.17)
3.5. PRELIMINARY SETTINGS AND DATA 95 Thus, the factor loading matrixΛsensor is set to the identity matrix I. That is, the usual method can be regarded as an estimation method by constraining Λsensor as I. In other words, the data rich method is a generalized estimation method, since it imposes only loose restrictions onΛsensor. Case SW (regular DSGE) assumes that the seven data indicators perfectly can be combined with seven model concepts. Let us denote the seven data indicators as the “primary indicators”Xsensor1,t . (2) Case A
Case A uses the same data in the case above, but the seven model concepts are assumed to be
“latent” variables due to imperfect observations, i.e., Case A introduces measurement errors. Note that we do not add the measurement error to nominal interest rate, since the central bank can perfectly control the nominal interest rate. This case can be considered as a benchmark case comparing the following two cases: The cases in a data rich environment with measurement errors.
We can express the measurement equation in Case A as follows:
Xsensor1,t = [I O]
S¯linkt S¯nont
+e1,t. (3.18)
where the primary indicatorsXsensor1,t is the same as those in Case SW ande1,t represents measure-ment errors.
(3) Case B
Case B corresponds to the data rich estimation method: We add seven new indicatorsXsensor2,t and each model concept is connected with two data indicators (“sensor” series). In other words, each model concept has two dependent variables if we regard the measurement equation as the regression model. The measurement equation in Case B can be described as:
Xsensor1,t Xsensor2,t
= I
Λsensor21 O O
S¯linkt S¯nont
+
e1,t e2,t
. (3.19)
where the matrixΛsensor is set as [I Λsensor21 ]0 and Λsensor21 is a diagonal matrix. In addition to the information ofXsensor1,t , ifXsensor2,t has useful information on the model concept, the factor loadings coefficient will be estimated as non-zero. e2,t denotes measurement errors corresponding to the additional indicators Xsensor2,t .
(4) Case C
Case C estimates the SW model by utilizing the full information of our entire data set. We further add seven indicators Xsensor3,t as the sensor series as does in Case B. Moreover, we introduce 34 additional indicators as the “information” series, Xinf ot . The relationship between the information series and the seven model concepts is not explicitly considered in the SW model. Thus, we cannot structurally interpret why this data and that model concept were connected. However, the extra information might bring estimation efficiency when extracting the common factors, i.e. estima-tion efficiency of model concepts (state variables). The measurement equaestima-tion in Case C can be represented as:
Xsensor1,t Xsensor2,t Xsensor3,t Xinf ot
=
I Λsensor21 Λsensor31 Λinf o1
O O O Λinf o2
S¯linkt S¯nont
+
e1,t
e2,t e3,t e4,t
. (3.20)
where the matrixΛsensor is set to [I,Λsensor21 ,Λsensor31 ]0, andΛsensor31 is a diagonal matrix. ButΛinf o1 and Λinf o2 are matrices with full elements. e3,t and e4,t are measurement errors corresponding to the indicators Xsensor3,t and Xinf ot , respectively. Following the measurement equation above, the information series does not exclude the possibility of connecting with all model concepts. However, from the burden of the computations, we imposeΛinf o2 =O, that is, the information seriesXinf ot is linked only to the observable states (the primary seven model concepts) ¯Slinkt . However, it still does not exclude the possibility that the information series can be linked to all seven model concepts.
It should be noted that, in Cases A, B, and C, we replaced the price and wage markup shocks, εpt and εwt, by the measurement errors, ept and ewt. The justification of this approach is two-fold:
First, inflation and wage data are known to be quite volatile in movements compared to other macroeconomic series. As pointed out by Stock and Watson (2007) and Edge and Gurkaynak (2010), especially recent inflation cannot be forecastable. Since we have already confirmed the fact in Chapter 2, we decide those variations should be captured not by markup shocks but by measurement errors. Justiniano and Primiceri (2008a) also adopts our strategy.
Second, the key difference between markup shocks and measurement errors is that the former being structural shocks and the latter being non-structural shocks. Suppose that the SW model still has misspecification problem and cannot explain all inflation and real wage variations. Nevertheless, estimating the SW model without measurement errors implies the structural shocks must replicate the volatile inflation and wage data fluctuations. In this case, biased structural shocks will be estimated. According to the state equation, structural shocks affect the model concept, and model concepts are interdependent with each other. Hence, the bias would spread all of the model concepts.
In particular, price and wage markup shocks directly undertake noisy data fluctuations. Moreover, when we let the markup shocks undertake data fluctuations, we cannot interpret their economic meanings structurally. Therefore, to avoid the negative spillover effect to the other latent variables (such as output gap), we filter out the volatile movements in the observed data by measurement errors. Replacing markup shocks with the measurement errors will lead to extract the unbiased common factors and focus on structural movements in inflation and wage data.
3.5.2 Data
The estimation period is, as is the same in Chapter 2, from 1981:Q1 to 1995:Q4 following Sugo and Ueda (2008) to exclude the periods during the zero interest rate policy.
We employ at most 55 quarterly Japanese macroeconomic series, consisting of 21 sensor series and 34 information series as summarized in Table 3.2 (a), (b). The seven primary indicatorsXsensor1,t are the same as in Chapter 2.
When selecting the additional data, we refer Bernanke and Boivin (2003) and Boivin and Gi-annoni (2006) that conducted the data rich estimation method in the U.S. We employed IIPs and price indices of various industries for output gap and inflation gap. Especially, since one of the aims is to estimate the monetary policy rule accurately and examine the effects of monetary policy shock, we utilize ten price indices consisting of three sensor series and seven information series. We also adopt the financial data as information series such as stock prices, money stocks and exchange rates. Again, the relationship between the information indicators and the seven model concepts is not certain. However, it is only because we do not make a structural model that takes into account those data, and in actual, the dynamics of the seven model concepts may be affected by those indicators: If the SW model is expanded to an open economy model, the exchange rate has a clear link with real interest rate (model concept) through the interest rate parity condition (through the FOC of foreign bond holdings and domestic bond holdings). If we incorporate the financial friction
3.6. RESULTS 97