Monthly, quarterly and annual growth rates
I received the following question by email:
I was wondering if mathematical relationships exist between monthly, quarterly, and annual growth rates (percentage changes). For time series data …
1) Is there a mathematical formula to know by how many months the Month-to-Month changes lead turns in this-month-to-the-same-month-one-year-ago changes (annual changes)?
2) Is there a mathematical formula to know by how many quarters the Quarter-to-Quarter changes lead turns in this-quarter-to-the-same-quarter-one-year-ago changes (annual changes)?
Also, regarding simple moving averages/totals:
3) Is there an exact or approximate formula to know by how many months a 3-month simple moving average (or moving total) leads a turn in a 12-month simple moving average (or moving total)?
As an example: Inflation is typically measured with Y/Y percentage changes in the CPI. So, it would be helpful to be able to identify when the 12-month rate of change will likely reach a top or bottom, using the 1-month or 3-months rates of change.
Monthly growth vs year-over-year growth
Suppose we have monthly data y_1,y_2,\dots. Then the monthly growth rates are given by g_t = \frac{y_t - y_{t - 1}}{y_{t - 1}}, and the year-over-year growth rates are given by G_t = \frac{y_t - y_{t - 12}}{y_{t - 12}}. A little algebra gives 1 + G_t = \prod_{j = 1}^{12} (1 + g_{t - j + 1}). \tag{1} So the year-over-year rate is the compounded product of the last 12 monthly rates. This gives a recurrence relationship: \frac{1 + G_t}{1 + G_{t - 1}} = \frac{1 + g_t}{1 + g_{t - 12}}. Therefore G_t > G_{t - 1} \iff g_t > g_{t - 12}. In other words, the year-over-year rate rises when the current monthly change exceeds the change 12 months ago, and falls when the reverse holds. Figure 1 shows why: moving from G_{t-1} to G_t, the 12-month window drops g_{t-12} and adds g_t, while the other 11 months are common to both.
For example, annual inflation peaks when the monthly rates entering the 12-month window fall below the rates dropping out of it. Because year-over-year growth is effectively a 12-month trailing window of monthly growth, month-to-month growth tends to lead year-over-year growth by about half the window length: the centre lag of an equally-weighted k-period trailing window is (k - 1)/2 periods, or 5.5 months when k = 12.
This centre-lag rule works best when the underlying series changes slowly within the 12-month window, as is typical of series like inflation, and is less reliable for more volatile or noisy series.
Quarterly growth vs year-over-year growth
The quarterly version is exactly analogous: just replace 12 by 4. Since four-quarter growth is a trailing 4-quarter window of quarterly growth, quarter-to-quarter growth leads year-over-year growth by about 1.5 quarters, under the same condition as above.
Log growth rates
Taking logarithms turns the multiplicative relationship above into a linear one, which is more convenient for compounding over several periods, for forecasting, and for series such as price indices, where growth is conventionally expressed in log form.
Define monthly log growth \ell_t = \ln(1 + g_t) and year-over-year log growth L_t = \ln(1 + G_t), using g_t and G_t from above. Taking logs of Equation 1 turns the product into a sum: L_t = \sum_{j = 0}^{11} \ell_{t - j}. So L_t rises when \ell_t > \ell_{t - 12}: the same turning condition as before, now linear rather than multiplicative.
This also gives a forecasting formula: L_{t + h} = L_t + \sum_{s = 1}^{h} \left(\ell_{t + s} - \ell_{t + s - 12}\right). So given a model for future monthly growth, you can forecast the future year-over-year rate in log terms.
The same idea works over any horizon, comparing the most recent 3-month block to the corresponding block a year earlier: L_t - L_{t - 3} = \sum_{j = 0}^{2} \ell_{t - j} - \sum_{j = 0}^{2} \ell_{t - 12 - j}. So the year-over-year rate has risen over the past three months if the most recent 3-month growth exceeds the same 3-month block a year earlier.
This is exactly how CPI inflation is usually reported: y_t is the price index, \ell_t the monthly inflation rate, and L_t the standard year-over-year rate quoted in the news.
3-month moving averages vs 12-month moving averages
A trailing simple moving average of \{y_1, y_2, \dots\} is defined as: \hat{T}_{k,t} = \frac{1}{k} \sum_{j = 0}^{k - 1} y_{t - j}. Then \hat{T}_{k,t} - \hat{T}_{k,t - 1} = \frac{1}{k} (y_{t} - y_{t - k}). So \hat{T}_{k,t} rises when the newest observation entering the window, y_{t}, exceeds the oldest observation leaving the window, y_{t - k}, and falls when the reverse holds.
As with the growth-rate case, the practical lead between two moving averages of different lengths can be approximated from the centre of each trailing window. A trailing k-period average is centred about (k - 1)/2 periods in the past, so a 3-month average is centred about 1 month in the past, while a 12-month average is centred about 5.5 months in the past. So a 3-month moving average tends to lead a 12-month moving average by about 5.5 - 1 = 4.5 months, for the same reason that monthly growth leads year-over-year growth above. As before, this holds as long as the series moves slowly relative to the 12-month window.