# How many measurement error would be best?

Dear Johhanes,

Could I ask two short questions?
1.When doing bayesian estimation, I see some paper add one measurement error(e.g. What’s news in business cycles); while some add many measurement errors (e.g. real business cycles in emerging countries). Could you help explain why?

1. When using Observable variable of “hours worked”, some paper use growth rate of h, while some paper use pecentage deviation from steady state. Does such difference matter?

Best regards,
Bo

Please read Pfeifer(2013): “A Guide to Specifying Observation Equations for the Estimation of DSGE Models” sites.google.com/site/pfeiferecon/Pfeifer_2013_Observation_Equations.pdf. Both topics are discussed there.

Many thanks. Sure, I have seen your guide for many times. But I am still not sure how many measurement errors I can add in measurement equations as **Exogenous variables,**since I do not see any example in any published paper. To make the shocks no less than observables, I need at least 2 measurement errors as exgoneous variables.
Is it O.K like:

[code]var_exo …y_Me c_Me i_Me h_Me;
model;

y_obs=y-y(-1)+d+log(dbar)+y_Me;
c_obs=c-c(-1)+d+log(dbar)+c_Me;
i_obs=i-i(-1)+d+log(dbar)+i_Me;
h_obs=h-h(-1)+h_Me;
end;

estimated_pramams;
stderr y_Me;
stderr c_Me;
stderr i_Me;
stderr h_Me;

end;
varobs y_obs c_obs i_obs h_obs;[/code]

Best regards,
Huan

What do you mean with as “exogenous variables”. As detailed in Chapter 6 of my Guide, using “Measurement Errors as a Special Case of Exogenous variables” is equivalent to regular measurement error using Dynare capabilities (i.e. in the observation equation of the Kalman filter). Thus, the papers you cite use that type of measurement error and what you try to do is done in the Garcia-Cicco et al (2010) AER paper and is a valid approach.

Dear all,

I read the paper in which authors estimate a DSGE model with measurement errors:
y_obs = y - y(-1) + epsilon_me_y.

The variance of the measurement errors is calibrated so that it corresponds to 10% of the variance in each data series.

After estimation authors report the following table (see file attached). My question is: how the numbers in the last column (i.e. structural explanation) are calculated?

Regards,
Measurement_error.pdf (47 KB)

By comparing the variance of the observed variable without measurement error

to the one with measurement error

Thank you for your answer. So, if I understand you correctly I need to compare theoretical variance (calculated at posterior mean, for example) when

var epsilon_me_y;
stderr 0;

and without this restriction?

Either that or you simply define

```y_growth=y - y(-1); y_obs=y_growth+epsilon_me_y;```
and then compare the variance of the two.

Thank you for your help. I augmented my model with (measurement) equations without measurement errors (denoted as _nome in table below) as you suggested. After doing so I run my model using stoch_simul command (using calibrated parameters) and I get the following:

THEORETICAL MOMENTS
VARIABLE MEAN STD. DEV. VARIANCE
pi_obs 0.0100 0.0090 0.0001
dy_obs 0.0060 0.0307 0.0009
dc_obs 0.0060 0.1556 0.0242
dg_obs 0.0060 0.0327 0.0011
di_obs 0.0060 0.0591 0.0035
dm_obs 0.0060 0.0180 0.0003
dx_obs 0.0060 0.0345 0.0012
pi_def_c_obs 0.0100 0.0088 0.0001
reer_obs 0.0000 0.1398 0.0196
dw_obs 0.0060 0.0079 0.0001
e_obs 0.0000 0.1024 0.0105
dyfor_obs 0.0060 0.0063 0.0000
pifor_obs 0.0100 0.0029 0.0000
rfor_obs 0.1226 0.0638 0.0041
r_obs 0.1226 0.0658 0.0043
pi_def_i_obs 0.0100 0.0081 0.0001
pi_obs_nome 0.0100 0.0090 0.0001
pi_def_c_obs_nome 0.0100 0.0088 0.0001
dy_obs_nome 0.0060 0.0307 0.0009
dc_obs_nome 0.0060 0.1556 0.0242
di_obs_nome 0.0060 0.0587 0.0034
dm_obs_nome 0.0060 0.0176 0.0003
dx_obs_nome 0.0060 0.0344 0.0012
dg_obs_nome 0.0060 0.0327 0.0011
reer_obs_nome 0.0000 0.1398 0.0196
e_obs_nome 0.0000 0.1023 0.0105
dw_obs_nome 0.0060 0.0079 0.0001
r_obs_nome 0.1226 0.0655 0.0043
pi_def_i_obs_nome 0.0100 0.0080 0.0001

There is no difference between std/variance of respective variable when there is measurement error and without measurement error.

P.S.: I calibrated the standard deviation of the measurement errors so that it corresponds to 10% of the standard deviation in each data series. For example, for output growth:

var epsilon_me_y
stderr 0.00118 = 0.10std(data)=0.100.0118;

To calculate the so-called structural explanation I specified:
y_growth=y - y(-1);
y_obs=y_growth+epsilon_me_y;

ect.

When estimating the model, the variables declared under varobs command are only those with measurement error, i.e. y_obs. Is this ok?

Should I declare also y_growth?

Yes, this is OK.
I finally found time to look at your initial issue. The reason is that stoch_simul currently ignores measurement error. Thus, the moments for the variables without measurement error are correct, but the ones with measurement error are understated.

If I understand you correctly, stoch_simul currently ignores all shocks that are included in an additive way?

My mistake. I overlooked that you specified the measurement error as a structural shock. stoch_simul is able to handle this. However, your measurement error hardly plays a role. When you look at the variance decomposition, they tend to explain 1% or less of the total variance. This will be hardly noticeable in the moments table.

I estimated my DSGE model with measurement error shocks. I included measurement errors in the model due to the stochastic singularity problem (despite having more shocks than observables in the model!). The standard deviation of the measurement error shocks is calibrated so that it corresponds to 10% of the standard deviation in each data series. After estimation I calculated historical decomposition and found that some observable variables (especially different measures of inflation, that is CPI inflation and domestic inflation based on GDP deflator) are driven mainly by measurement error shock (figure attached). How can these results be interpreted?

P.S. The measurement equation for inflation is the following:
pi_obs = (pi_ss - 1) + pi_hat + epsilon_me.

The historical decomposition is based on pi_obs. Should I calculate historical decomposition based on pi_hat?
HistDecom.pdf (154 KB)

The picture is a bit strange, because it looks as if the measurement error accounts for more than 10% of the volatility.

Generally, a picture like this tells you that (believing that we do not mismeasure inflation that badly) your model is misspecified and thus has a hard time accounting for the movement of the observables with other types of shocks.