# Observation equations

Dear all,

I have got a question regarding the specification of observation equations.

My model at hand has a period length of a quarter. I entered my model equations non-linearly into the model-block. However, for all endogenous variables in levels, I used the exp()-substitution in the model-block. I did not use the exp()-substitution for net interest rates and all other variables measured in percentage.

I gathered the time series below:
i) GDP at current prices (quarterly)
ii) GDP deflator (quarterly)
iii) Employment and Unemployment (quarterly, in levels)
iv) HICP index (quarterly)
v) Debt to GDP ratio (quarterly)
vi) Gross fixed capital formation (quarterly)
vii) Euribor1W (quarterly)

Data preparation:
i) Target: Percentage deviations of GDP from it’s trend
To obtain a time series of real GDP, I divided the nominal GDP values by the GDP deflator values. Afterwards, I divided real GDP by the active labor force (employment + unemployment) to obtain real GDP per capita. Afterwards, I applied the logarithm and used a log-quadratic trend to extract the cyclical component as percentage deviations from trend. Finally, I demeaned the data series.

ii) Target: Gross fixed capital formation as a percentage of GDP
I divided the gross fixed capital formation values by (employment + unemployment). Afterwards, I divided the resulting values by real GDP per capita to obtain gross fixed capital formation as a percentage of real GDP per capita.

iii) Target: Net inflation rate
First, I applied the logarithm to HICP index. Then, I calculated log(HICP_{t}) - log(HICP_{t-1}) to obtain values for the net inflation rate.

iv) Target: Quarterly net interest rate
I used the Euribor1W to calculate the quarterly net interest rate by

i_{quart} = (1+Euribor1W)^{1/4} - 1

v) Target: Debt to GDP ratio
I did not manipulate the gathered data series for the debt to GDP ratio.

Next, I specified the following five observation equations:

// observation equations
percentageDeviationOutput_obs = Y_tilde - steady_state(Y_tilde) + sig_y * eps_y;
i_obs = i;
inflation_obs = pii;
debtGDPRatio_obs = 1/4 * (exp(B_tilde)/exp(Y_tilde));
GFCFAsPercentageOfGDP_obs = (exp(K_tilde)-exp(K_tilde(-1)))/exp(Y_tilde);

where:
pii denotes the net inflation rate.
B is the stock of government debt.

When I use the first two observation equations (DY_obs and i_obs), the estimation works. However, when I include all of them, I get the following error message:

„POSTERIOR KERNEL OPTIMIZATION PROBLEM!
(minus) the hessian matrix at the “mode” is not positive definite!
=> posterior variance of the estimated parameters are not positive.“

I also tried another optimization routine („mode_compute = 6“ and „mode_compute = 9“), which unfortunately did not solve the issue. Therefore, I presume that I’ve most likely misspecified my observation equations.

Could someone give me a hint where I have done something wrong?

Thank you!

Best,
Matthias

1 Like

Does the estimation work with simulated data? Thus, simulate data from a calibrated version of the model and use the simulated data to estimate the model. I do that sometimes to be sure the problem is not with the observed data. Maybe you could also attach a picture of your observed data…

Without the files it is impossible to tell. What do the `mode_check`-plots show?

Thank you both very much for your help. The mode_check-plots indicated some prior bound issues. When looking at the simulated data, I noticed that the time series did not yet match the model. Now, I have solved this and the prior bound issue disappeared. Thanks again!