GDP (Quarterly) Prediction in Mixed Frequency DFM

[SebKrantz]

Dear Chad, thanks a lot for the excellent work of implementing these codes in Python. Regarding your example given in this notebook, you compute the quarterly forecast / nowcast from its unobserved monthly counterpart using

# Resample to quarterly frequency by taking the value in the last
# month of each quarter
point_forecasts_q = point_forecasts_m.resample('Q').last()

However, this does not correspond to the internal modelling a la Mariano and Murwasa (2003) or Banbura and Modugno (2014), where the quarterly series growth rate is a weighted average of the unobserved monthly series. I summarized this in this note here:

So, if you follow Banbura and Modugno (2014) as I presume, should not the quarterly forecast be obtained as:

def wma_BM(x):
    return x + 2*x.shift(1) + 3*x.shift(2) + 2*x.shift(3) + x.shift(4)

wma_BM(gdp_forecast)

Final comment on the notebook: the FRED databases are no longer on amazon AWS, the base URL should be changed to:

base_url = 'https://files.stlouisfed.org/files/htdocs/fred-md'

[ChadFulton]

Thanks Sebastian, glad to hear these have been helpful for you!

As long as you specify the quarterly variables using the endog_quarterly argument, then the temporal aggregation structure that you're referring to is built-in to the state space model (accounting for this feature is why the endog_quarterly value exists).

For monthly variables, the observation equation has the form:

$$y_{it}^M = \lambda_i ft + \varepsilon{it}$$

while for quarterly variables, it has the form:

$$y_{it}^Q = (\lambda_i ft + \varepsilon{it}) + 2 (\lambdai f{t-1} + \varepsilon_{it-1}) + 3 (\lambdai f{t-2} + \varepsilon_{it-2}) + 2 (\lambdai f{t-3} + \varepsilon_{it-3}) + (\lambdai f{t-4} + \varepsilon_{it-4})$$

You can confirm this by examining the "design" matrix $Z_t$ of the observation equation:

$$y_t = Z \alpha_t + \varepsilon_t$$

which you can find as results.model['design'].

Final comment on the notebook: the FRED databases are no longer on amazon AWS, the base URL should be changed to:

Thanks for letting me know.

[SebKrantz]

Thanks a lot for the swift response and clarification! My confusion was because the prediction of GDP is a monthly prediction, which made me think you are just returning $\tilde{y}_{it}^m = \lambda_i ft+\epsilon{it}$. I get now from your response that it is a monthly prediction of quarterly growth rates that is being returned. Since this is not very intuitive, it would be good to state this in the documentation for the DynamicFactorMQ class.

[ChadFulton]

Thanks, that is a good idea to note this in the documentation.