I have used your dynare codes for Basu and Bundick (2017) and constructed impulse response function starting at the stochastic steady state /Ergodic mean in the absence of shocks(EMAS) with stationary productivity shocks. I am using 3rd order approximation.

In the second moments volatility of consumption is greater than volatility of output. IRFs depicts greater initial response in consumption compared to output on impact after a positive TFP shocks. So this is consistent with second moments.

The intuition is that as the household now take into account the likelihood of positive TFP shocks (which actually do not happen) as they expect more income, hence, they response to it by increasing consumption. Is this intuition correct?

What exactly is your shock experiment? You description does not sound like it depends on Jensen’s Inequality as you would expect for second moment shocks

Thanks for the prompt response. I am calibrating the model to emerging countries data. When I use the built in IRF option at 3rd order. Then, we don’t observe greater initial response in consumption compare to output after positive TFP shocks which contradicts the second moments of models. That,s why I opted for IRF at EMAS.
Sorry, I could not get your point. What exactly I should look for?. I have given the codes below. Would you please refer to these codes to explain the problem as you mention.? Is there a way to confirm from the model whether Jensen’s Inequality holds or not? In our case 3rd derivative of utility(GHH) is positive. Which means a rise in uncertainty of about future income raise saving and decrease consumption. But this saving and consumption must depend on others parameters values such as relative risk aversion etc. But Isn’t the case with IRFs at EMAS that household now expecting positive income in future(positive TFP) and they respond to it by raising consumption in beginning?

sigmae = 0.0201;
rho = 0.86;
a = rho*a(-1)+ sigmae*e;
var e; stderr 1;
y_pos=strmatch('y',M_.endo_names,'exact');
c_pos=strmatch('c',M_.endo_names,'exact');
i_pos=strmatch('i',M_.endo_names,'exact');
h_pos=strmatch('h',M_.endo_names,'exact');
tb_y_pos=strmatch('tb',M_.endo_names,'exact');
ca_y_pos=strmatch('ca',M_.endo_names,'exact');
Z_pos=strmatch('a',M_.endo_names,'exact');
d_pos=strmatch('d',M_.endo_names,'exact');
lambda_pos=strmatch('lambda',M_.endo_names,'exact');
IRF_periods=50;
burnin=5000; %periods for convergence
shock_mat_with_zeros=zeros(burnin+IRF_periods,M_.exo_nbr); %shocks set to 0 to simulate without uncertainty
IRF_no_shock_mat = simult_(oo_.dr.ys,oo_.dr,shock_mat_with_zeros,options_.order)'; %simulate series
stochastic_steady_state=IRF_no_shock_mat(1+burnin,:); % stochastic_steady_state/EMAS is any of the final points after burnin
shock_mat = zeros(burnin+IRF_periods,M_.exo_nbr);
shock_mat(1+burnin,strmatch('e',M_.exo_names,'exact'))= 1;
IRF_mat = simult_(oo_.dr.ys,oo_.dr,shock_mat,options_.order)';
IRF_mat_percent_from_SSS = (IRF_mat(1+burnin+1:1+burnin+IRF_periods,:)-IRF_no_shock_mat(1+burnin+1:1+burnin+IRF_periods,:))./repmat(stochastic_steady_state,IRF_periods,1); %only valid for variables not yet logged
%scale IRFs as reqired
y_vola_IRF = 100*IRF_mat_percent_from_SSS(:,y_pos);
c_vola_IRF = 100*IRF_mat_percent_from_SSS(:,c_pos);

All I was saying is that HH at the EMAS do not expect future positive TFP (which would imply a level shift, i.e. a first moment movement), but factor in the occurrence of future shocks, both positive and negative. They take into account the uncertainty/variance, which is a second moment movement.

Thanks a lot! I was ignoring negative future shocks. It is uncertainty that HH take into account. I have applied to EMAS to Seoane, published at JEDC by replicating his second moments. But now IRFs at EMAS can’t show greater initial impact in consumption compare to output. In his paper second moments though, volatility of consumption is greater than the output volatility. I infer from it. It is not IRFs at EMAS that can show greater initial impact in consumption compare to output rather the values of other parameter that matters a lot.

I am not sure I get the point. There is not an immediate mapping from IRFs to (potentially filtered) second moments. What matters is impact response and persistence (and filtering)

Exactly, this is what I meant when I said in previous notes, it depends on the values of the parameters.(What matters is impact response and persistence") For example higher gamma (risk aversion) can bring more persistence in consumption. " though at the higher value of gamma initial response in consumption shall be smaller after a positive TFP.

Please find attached TFP shocks IRF at EMAS.pdf (122.1 KB) for EDF,IDF,PAC,EDEIR and IDEIR, IRF at EMAS and GIRF. As you mentioned in previous note that HH at the EMAS take into account uncertainty(could be positive TFP or negative TFP). I thought to dig it a bit deeper. I am using 3rd order approximation.

IRF at EMAS for TFP is positive for IDF,EDF and PAC models. While it is negative in EDEIR and IDEIR model in beginning then it gets positive. Whereas, GIRF for TFP is same for the all models. I am trying to understand institution behind these difference IRF movements.

We know that the dynamics of closing devices of Schmitt-Grohé and Uribe (2003) are internally very different.In IDF and EDF models, discount factor is decreasing in consumption by assumption a positive TFP shocks increase the lifetime wealth, in turn, results in an increase in current consumption, which reduces discount factor, so this make household impatient, and they further increase their current consumption.

My first question is why IRF at EMAS for TFP in IDF and EDF models are positive? Is it because, domestic interest rate is constant. Dynamic arising from HH debt market does not spread.HH is not aware the consequences of his borrowing so the take TFP as positive.

Now, IDEIR and EDEIR models real interest rate is increasing in foreign debt by construction. so this higher interest rate will have dampening effects on consumption in next round. My second question is why IRF at EMAS for TFP in EDEIR and IDEIR models are negative in the beginning and the positive?*

Is it because, as risk premium and r goes up by construction. Now HH expect TFP as negative.

Last, question is why PAC model IRF at EMAS are postive.

Sorry, but those a very deep questions I am unable to answer. The answers are far from obvious. They would require a thorough model investigation I am unable to conduct given my time budget.