Corrected exercises: time series and forecasting (360-301)

The TREND is the underlying long-term movement, here the growth in enrolment. The CYCLE is an oscillation lasting several years, tied to the economic climate, whose period is not fixed. SEASONALITY is a pattern with a FIXED, known period, here four quarters. The IRREGULAR component is what is left over, unpredictable: a snowstorm, a strike, a change in regulations.

The trend shows in the annual means: $\dfrac{2100+1500+2400+1800}{4}=1950$, then 2040, then 2140, a steady rise of about 95 students per year. The seasonality shows in the shape repeated within each year: the third quarter is always the highest, around 2400 to 2640, the second always the lowest, around 1500 to 1640, and the pattern recurs in all three years.

Both are oscillations, but SEASONALITY has a FIXED period set by the calendar, and its pattern repeats almost identically: fall term, winter term, summer term. The CYCLE has a VARIABLE period, from a few years to ten or so, and an irregular amplitude: a recession pushes CEGEP enrolment up, a recovery brings it down, with no way of saying in advance when. That is why we routinely deseasonalize and never “decycle”: the cycle cannot be predicted.

MULTIPLICATIVE model: $Y_{t}=T_{t}\times S_{t}\times C_{t}\times I_{t}$, where the seasonal effect is a percentage of the trend. ADDITIVE model: $Y_{t}=T_{t}+S_{t}+C_{t}+I_{t}$, where the seasonal effect is a gap in units. The multiplicative model fits here: the gap between the high quarter and the low quarter goes from 900 in the first year to 1000 in the third, so it grows with the level of the series. A seasonal gap that is proportional calls for the multiplicative model; a constant gap would call for the additive one.

Because the two quarters do not carry the same seasonal weight: the third is structurally high and the fourth structurally average, so a drop between them is NORMAL and says nothing about the health of the institution. Comparing two consecutive quarters only makes sense after deseasonalization. The usual alternative in the financial press is to compare a quarter with the SAME quarter of the previous year, which neutralizes the season but costs twelve months of responsiveness.

First four quarters: $\dfrac{2100+1500+2400+1800}{4}=\dfrac{7800}{4}=1950$. Quarters 2 to 5: $\dfrac{1500+2400+1800+2200}{4}=\dfrac{7900}{4}=1975$. The smoothed series rises gently, from 1950 to 1975, whereas the raw series went from 1500 to 2200 over the same quarters: all the seasonal movement has disappeared.

Because each window of four quarters contains each season exactly ONCE: one fall term, one winter term, one summer term, one intersession. The sum of the seasonal effects is therefore the same from one window to the next, and it cancels in the comparison. With any other order, the windows would not contain the same seasons and the smoothing would leave a seasonal residue. For monthly data, you would use order 12.

You lose two values at each end, four in all out of twelve: a moving average of even order $p$ loses $\dfrac{p}{2}$ values on each side. Eight smoothed points remain. This is a major drawback in practice: the moving average is blind precisely to the most recent period, the one the decision-maker cares about.

The mean of quarters 1 to 4 belongs to the middle of the window, that is, position 2.5, which matches no quarter. You CENTRE it by averaging two consecutive moving averages: $\dfrac{1950+1975}{2}=1962.5$ then belongs to quarter 3. This is the centred moving average of order 4, the one every statistical agency uses on quarterly data.

A moving average of order 3 would mix three seasons out of four and leave a good part of the seasonal pattern in place: the smoothed series would still oscillate. On our data, the window of quarters 1 to 3 gives $\dfrac{2100+1500+2400}{3}=2000$ and the window of quarters 2 to 4 gives $\dfrac{1500+2400+1800}{3}=1900$, a gap of 100 that has nothing to do with a trend: it is a leftover of the season. The rule has no exceptions: the order of the smoothing must be a MULTIPLE of the seasonal period.

Mean of year 1: 1950. Ratios: $\dfrac{2100}{1950}\approx 1.0769$; $\dfrac{1500}{1950}\approx 0.7692$; $\dfrac{2400}{1950}\approx 1.2308$; $\dfrac{1800}{1950}\approx 0.9231$. Their sum is exactly 4, which is automatic since each value is divided by the mean of the four.

First quarter: $\dfrac{1.0769+1.0784+1.0841}{3}\approx 1.0798$. Repeating the same calculations for the other quarters: second, $\approx 0.7668$; third, $\approx 1.2332$; fourth, $\approx 0.9202$. Averaging over the three years removes most of the irregular component: that is the point of this second step.

$1.0798+0.7668+1.2332+0.9202=4.0000$. It must equal 4 because the indices are ratios to the mean: if their sum exceeded 4, the season would add level to the series, which is absurd. When the sum departs from 4 because of rounding or the irregular component, you RENORMALIZE by multiplying each index by $\dfrac{4}{\text{sum obtained}}$.

An index of 0.7668 means that the second quarter sits structurally 23.3% BELOW the average level of its year. This is not a drop, it is a permanent feature of the academic calendar, and it will recur next year. A release announcing a “29% plunge in enrolment in the second quarter” would be describing the calendar, not the institution.

Second quarter of year 1: $\dfrac{1500}{0.7668}\approx 1956$. Third quarter of year 3: $\dfrac{2640}{1.2332}\approx 2141$. The deseasonalized values are thus close to the respective annual means, 1950 and 2140, which is the check to make: once the season is removed, only the trend and a little irregular variation should remain.

$\overline{x}=2$ and $\overline{y}=\dfrac{1950+2040+2140}{3}=\dfrac{6130}{3}\approx 2043.33$. Covariance sum: $\left(-1\right)\left(1950-2043.33\right)+0+\left(1\right)\left(2140-2043.33\right)=93.33+96.67=190$. Sum of squares of the $x$ deviations: $1+0+1=2$. Slope: $\dfrac{190}{2}=95$. Intercept: $2043.33-95\times 2=1853.33$. Line: $\hat{y}=1853.33+95x$.

Year 4: $\hat{y}=1853.33+95\times 4=1853.33+380=2233.33$ students per quarter on average.

Multiply the trend by each seasonal index. First quarter: $2233.33\times 1.0798\approx 2412$. Second: $2233.33\times 0.7668\approx 1713$. Third: $2233.33\times 1.2332\approx 2754$. Fourth: $2233.33\times 0.9202\approx 2055$.

Sum of the four forecasts: $2412+1713+2754+2055=8934$, and $4\times 2233.33=8933.33$. The agreement, up to rounding, is automatic since the indices add up to 4. It is the final check on any seasonal forecast, and it catches index errors.

Three reservations. First, the line is fitted on only THREE points: the trend is very poorly estimated, and there is only one degree of freedom left to judge the fit. Second, extending a trend assumes that nothing changes, which rules out by construction any break, reform, program closure or demographic shock; yet these breaks are precisely what the decision-maker cares about. Third, no uncertainty is quantified here: a serious forecast comes with an interval, and that interval widens quickly as you move away from the data. A one-year forecast is reasonable; at five years it is decorative.

Two intervals separate the first and the third year. $\left(1+t\right)^{2}=\dfrac{2140}{1950}\approx 1.09744$, so $1+t=\sqrt{1.09744}\approx 1.04759$ and $t\approx 4.76\%$ per year. Note that the total rise of 9.74% divided by 2 would give 4.87%, a little too high: the geometric mean is always smaller than the arithmetic mean.

LINEAR model: $1853.33+95\times 10=2803.33$. EXPONENTIAL model: starting from 1950 in year 1, $1950\times 1.04759^{9}\approx 1950\times 1.5203\approx 2965$. The gap thus reaches 162 students in the tenth year, whereas the two models were indistinguishable over the three observed years. This is the fundamental problem of extrapolation: two models that fit the past equally well diverge in the future, and nothing in the data allows you to choose between them.

A STRAIGHT LINE. A constant rate means that the ratio of one value to the next is constant, so the DIFFERENCE of the logarithms is too: $\ln\left(y_{t+1}\right)-\ln\left(y_{t}\right)=\ln\left(1+t\right)$. On a logarithmic scale the series therefore becomes a straight line, and the slope of the line directly measures the growth rate. This is why long economic series, such as gross domestic product and population, are almost always published this way.

The series with a constant rate always ends up overtaking the other, whatever its rate and whatever the slope of the other. Starting both from 100: after 10 years, $100+100=200$ against $100\times 1.03^{10}\approx 134$; after 40 years, $500$ against $100\times 1.03^{40}\approx 326$; after 80 years, $900$ against $100\times 1.03^{80}\approx 1064$. The crossover happens around $t\approx 70.6$, during the seventy-first year: $100\times 1.03^{70}\approx 792<800$, but $100\times 1.03^{71}\approx 816>810$. Exponential growth is slow at first and unbounded afterwards, which is exactly why human intuition always underestimates it.

By the curvature: the successive gaps increase instead of staying constant. The quick test is to compute the DIFFERENCES from one value to the next; if they are roughly constant, growth is linear, and if it is the RATIOS that are constant, growth is exponential. On our annual means, the differences are 90 then 100, and the ratios 1.0462 then 1.0490: both are nearly constant, which confirms that with three points you cannot decide.

From 1560 to 2520: $\dfrac{2520-1560}{1560}\approx 0.6154$, a rise of 61.5%. The number is exact and it measures little more than the passage from winter to fall.

Deseasonalized values: $\dfrac{1560}{0.7668}\approx 2034.4$ and $\dfrac{2520}{1.2332}\approx 2043.5$. Change: $\dfrac{2043.5-2034.4}{2034.4}\approx 0.0045$, that is 0.45%. The real movement is therefore under half a percent, against 61.5% in appearance: all the rest was calendar.

That the figure describes the season and not the institution. A “61% jump” will recur next year and the year after, whatever happens, and it would be just as true at a CEGEP in decline. The correct release would say: “after adjustment for seasonal variation, enrolment is up 0.4% from the previous quarter”, or failing that it would compare with the same quarter of the previous year.

Third quarter of year 2 against third quarter of year 1: $\dfrac{2520-2400}{2400}=0.05$, that is 5.0%. The method is simple and robust, and it needs no seasonal index. It costs two things: RESPONSIVENESS, since a turnaround that happened three months ago stays invisible for nine months, and SENSITIVITY to accidents, since an exceptional quarter a year ago distorts today's comparison.

Its main limitation is that deseasonalization is an ESTIMATE, revised with every new data point: the seasonal indices are recomputed, and the value published for a past quarter changes retroactively. Statistical agencies do publish these revisions, often a few tenths of a point, which is enough to make a “technical recession”, defined as two consecutive quarters of decline, appear or disappear. In addition, a deseasonalized series is NOISIER than it looks: by removing the most regular component, you leave the irregular component in the foreground, and a single month can be enough to create a false inflection.

AUTOCORRELATION is the correlation of a series with itself shifted in time: this quarter's enrolment resembles the previous quarter's, and resembles even more the same quarter of last year. Example: in our data, the autocorrelation of order 4, the one linking each quarter to the same quarter of the previous year, is close to 1, since the seasonal pattern repeats almost identically.

The tests based on the INDEPENDENCE of the observations, starting with the standard errors of the regression and all the significance tests that follow from them. They almost always go wrong in the same direction: they UNDERESTIMATE the standard errors, hence overestimate the $t$ statistics, hence declare significant relationships that are not. The mechanism is easy to grasp: forty quarters strongly correlated with one another contain far less information than forty independent observations, and the formula behaves as if they contained as much.

It is a SPURIOUS TREND CORRELATION, or spurious regression: any two series that rise are correlated, because they share time as a common factor. The remedy is to work on the CHANGES rather than the levels: you compute the differences from one period to the next, or the growth rates, and correlate those. Most of the time, a correlation of 0.97 on the levels falls to almost zero on the changes, which settles the question.

You must check whether the two series simply share the same SEASON. Two monthly series that both peak in July will be strongly correlated with no link at all: ice cream sales and drownings, electricity consumption and colds. So you deseasonalize both before computing any coefficient. Twelve monthly points, for that matter, allow no conclusion, whatever the value obtained.

Much smaller than 40. The effective size is computed by $n_{eff}\approx n\dfrac{1-\rho}{1+\rho}$, where $\rho$ is the autocorrelation of order 1: with $\rho=0.8$, you get $40\times \dfrac{0.2}{1.8}\approx 4.4$ effective observations. Forty strongly autocorrelated quarters are therefore worth, in information, fewer than five independent observations. It is the same reasoning as the design effect in cluster sampling, and it explains why economic series require methods of their own.

FALSE. The 61% rise is a CALENDAR effect, structural and repeated every year: the seasonal index of winter is 0.767 and that of fall is 1.233. After adjustment, the real movement is 0.45%. Correct statement: seasonally adjusted, enrolment is up 0.4% from the previous quarter, which is too small to be attributed to anything.

FALSE. The order of the smoothing must be a MULTIPLE of the seasonal period, here 4, otherwise each window contains a different combination of seasons and the smoothing leaves the pattern in place. On our data, a moving average of order 3 still oscillates by 100 units from one window to the next. Correct statement: use order 4 on quarterly data and order 12 on monthly data, even if it means losing points at the ends.

FALSE. Any two growing series are strongly correlated because they share time: this is the spurious trend correlation, and it produces coefficients above 0.9 between quantities with no connection whatsoever. Correct statement: you must correlate the CHANGES from one period to the next, not the levels; the correlation computed that way is almost always close to zero.

FALSE. The line was fitted on three points and three years: extending it by thirty years multiplies the horizon of the data by ten, with no reason to believe that the trend will hold. In addition, the uncertainty of a forecast grows with distance, and no interval is given. Correct statement: the observed trend is about 95 students a year; a forecast one or two years out is defensible, beyond that it is a scenario and not statistics.

FALSE, you must RENORMALIZE. A sum of 4.08 means that the indices add 2% of level to the series, which would inflate every forecast and make the final check fail. Correct statement: multiply each index by $\dfrac{4}{4.08}\approx 0.9804$, so that the sum becomes exactly 4 again.

Year 1: $\dfrac{840+610+960+730}{4}=\dfrac{3140}{4}=785$. Year 2: $\dfrac{900+640+1020+780}{4}=\dfrac{3340}{4}=835$. Year 3: $\dfrac{950+690+1090+830}{4}=\dfrac{3560}{4}=890$.

Ratios, year 1: $1.0701$; $0.7771$; $1.2229$; $0.9299$. Year 2: $1.0778$; $0.7665$; $1.2216$; $0.9341$. Year 3: $1.0674$; $0.7753$; $1.2247$; $0.9326$. Averages by quarter: $1.0718$; $0.7730$; $1.2231$; $0.9322$. Sum: $4.0001$, that is 4 up to rounding, which validates the calculation.

$\overline{x}=2$, $\overline{y}=\dfrac{785+835+890}{3}=\dfrac{2510}{3}\approx 836.67$. Covariance sum: $\left(-1\right)\left(785-836.67\right)+\left(1\right)\left(890-836.67\right)=51.67+53.33=105$. Sum of squares of the deviations: 2. Slope: $52.5$. Intercept: $836.67-105=731.67$. Line: $\hat{y}=731.67+52.5x$.

Trend for year 4: $731.67+52.5\times 4=731.67+210=941.67$ visits per quarter on average. Forecasts: $941.67\times 1.0718\approx 1009$; $941.67\times 0.7730\approx 728$; $941.67\times 1.2231\approx 1152$; $941.67\times 0.9322\approx 878$. Check: their sum is about 3767, that is $4\times 941.67$.

First, you must say that the second quarter is structurally the SLOWEST, with an index of 0.773, that is 23% below the annual mean: hiring for that quarter would mean hiring for the least busy period. Next, the real question concerns the busiest quarter, the third, whose forecast goes from 1090 to about 1152 visits, 62 more. Finally, the answer must come with its limits: the line rests on three points, no interval is computed, and the opening of a program or a change in schedule would be enough to invalidate the forecast. The reasonable recommendation is to adjust resources for quarter three, and to follow the deseasonalized series quarter after quarter rather than lock in a plan based on an extrapolation.

Barely. With a margin of 0.2 points, a decline of 0.3 points leaves an interval from $-0.5$ to $-0.1$ points: it does not contain zero, so the decline is distinguishable, but only just. And the comparison with the previous quarter, itself affected by the same uncertainty, is even more fragile: the difference between the two, $0.1$ then $-0.3$, a gap of 0.4 points, has a margin of about $0.2\sqrt{2}\approx 0.28$ points, which makes it only just significant. A movement of this size should never make a headline.

The QUARTERLY change is responsive but noisy: it picks up a turnaround as soon as it happens, at the price of a mediocre signal-to-noise ratio. The TWELVE-MONTH change is stable but slow: here, 1.4%, it describes the year just ended and will say nothing about a turnaround that occurred three months ago until several quarters have passed. A careful reader looks at both, and becomes wary when they contradict each other, which is precisely the case here: employment is up over a year and down over a quarter, which at best signals a recent slowdown.

Because without seasonal adjustment, comparing a second quarter with a first quarter would be meaningless: employment follows a very pronounced calendar, with student hiring, seasonal work and the end of school contracts. Raw seasonal swings are counted in percentage points, ten times the change being reported. Without adjustment, the release would be describing spring.

It relies on a THRESHOLD comparison: saying that a decline is the first in two years suggests a remarkable event, whereas over eight quarters, seven of them tiny rises, it only takes one change going from $+0.1$ to $-0.3$ to produce that sentence. It is a narrative dressing of a movement consistent with noise. To judge, you would need the full series of the eight changes with their margins, and not the single qualifier “first”.

“Seasonally adjusted employment fell 0.3% in the second quarter, after a 0.1% rise in the first. Given the survey's margin of error, plus or minus 0.2 points, this decline is only weakly established, and the difference between the two quarters is at the limit of what the survey can distinguish. Over twelve months, employment is up 1.4%. These figures are therefore consistent with a recent slowdown, without a turnaround having been demonstrated; one or two more quarters will be needed to settle the matter.” The paragraph keeps the facts, adds the uncertainty to every number, contrasts the two comparison horizons, and replaces the announcement of an event with a hypothesis to be confirmed.