Quarterly Debt Steady-State

Hi everyone,

I have been trying to calibrate my model matching the average debt to GDP ratio. However, I only have annual data on the debt and my model is on a quarterly basis.

I have observed that in annual and quarterly data the same ratios are not always preserved, for example:

If we take the GFKF-to-GDP ratio as the steady-state value of the investment, the annual and quarterly value does not vary much, however,

K_{ss}^{Y}=\frac{I_{ss}}{\delta^Y}\neq K_{ss}^{Q}=\frac{I_{ss}}{\delta^Q}

This given that the annual capital depreciation rate is four times the quarterly depreciation rate.
Therefore, the steady-state value of capital on a quarterly basis is

K_{ss}^{Q} = 4\times K_{ss}^{Y} .

My question is: will the same happen with quarterly debt? This given that something similar happens with the interest rate and the discount factor.

Debt is a stock, GDP is a flow. If you compute the annual debt to GDP ratio, you are dividing the debt stock at a point in time by the annual GDP. Annual GDP is the sum of quarterly GDPs. Thus, the annual debt to GDP ratio is just one quarter of the quarterly one.

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Thank you very much; I was confused because I found quarterly information on the debt position, and it had a similar magnitude to the annual debt, but I see that the position is different from the stock.
However, I would like to add that information as an observable variable of my model. If I have the following expression:
G_{t} + (1+r_{t-1})D_{t-1}\leq TR_{t}+D_{t}
How can I define the observable of the public debt position?

The debt stock is not affected by time aggregation as you are measuring it at a point in time. That is not the problem. The issue is the scaling. So ideally, you observe the quarterly debt to GDP-ratio. The mapping should be straightforward. In the data, you divide debt by the GDP in that quarter (be careful that it is not annualized). Then in the model, you can map it to D/Y at quarterly frequency.

The biggest problem usually is to make the government budget constraint hold with the average level of debt observed in the data. It is often inconsistent with the average tax levels observed in the data. So you may need to adjust lump sum taxes. See e.g. Born/Peter/Pfeifer (2013): Fiscal news and macroeconomic volatility, https://doi.org/10.1016/j.jedc.2013.06.011

That was my mistake; I was calculating the ratio with the annualized quarterly GDP. I looked for the de-annualized GDP, and everything matches; the annual average debt to GDP ratio is approximately a quarter of the quarterly debt to GDP ratio.

Again, thank you very much.