Multi-Step Forecasts From A Smoothed Level

Why Multi-Step Forecasts Stay Flat

Simple exponential smoothing (SES) uses a current level to summarize the information in past observations. Once that level has been calculated, it is the point forecast for the next period. The important multi-step result is that the forecast remains at that same level for every future horizon, provided no new observation has arrived.

A flat sequence of forecasts does not mean future observations must be equal. Rather, SES has no separate trend or seasonal component that would move its point forecast as the horizon increases. As new observations become available, the level is updated, and forecasts made from that new time point can change.

Definition: Let $y_t$ be the observed value at time $t$, and let $\ell_t$ be the smoothed level after observing $y_t$. In simple exponential smoothing, for a smoothing parameter $0<\alpha\leq 1$, $$ \ell_t=\alpha y_t+(1-\alpha)\ell_{t-1}. $$ Given observations through time $T$, the SES point forecast $h$ periods ahead is denoted by $\hat{y}_{T+h|T}$. Under the standard SES forecasting assumptions, $\hat{y}_{T+h|T}=\ell_T$ for every integer $h\geq 1$.

The initial level $\ell_0$, or another chosen starting level, is needed to begin the recursion. Its selection is separate from the multi-step forecasting rule: once the level at time $T$ is available, all forecasts from that origin use $\ell_T$. The value of $\alpha$ determines how much a newly observed value changes the level, but it does not make the forecast slope upward or downward as the horizon grows.

Forecasting From a Fixed Origin

A forecast origin is the time at which the available data stop. For example, a forecast made at origin $T$ can use $y_1,\ldots,y_T$, but not an observation from $T+1$. The one-step forecast is the current level:

\hat{y}_{T+1|T}=\ell_T.

For a longer horizon, the forecast procedure treats each future observation as unknown. In the standard SES model, the expected future observation equals the expected future level. Since the unknown future observation has no expected error away from that level, the expected level does not move either. The result is the same point forecast at $h=2$, $h=3$, and every later horizon.

This statement is about the forecasts calculated from one fixed origin. It does not say that every later forecast, after new data arrive, must stay unchanged. A newly observed $y_{T+1}$ enters the level update and can shift all forecasts calculated from time $T+1$.

Model assumption: A standard innovations representation of SES is $y_t=\ell_{t-1}+\varepsilon_t$ and $\ell_t=\ell_{t-1}+\alpha\varepsilon_t$, where future innovations have conditional mean zero given the information available at the forecast origin. The flat multi-step result is a conditional-mean forecast under this assumption. It does not assert that future observations equal the forecast, and it need not describe a process with an unmodeled trend or seasonal pattern.

Why the Multi-Step Formula Holds

The result can be justified by recursively forecasting the level. At time $T$, the current level $\ell_T$ is known. The next observation is forecast at that level. To forecast the level one period beyond $T$, use the SES update with the forecasted observation in place of an unavailable actual observation. This makes no adjustment to the level:

\hat{\ell}_{T+1|T} =\alpha\hat{y}_{T+1|T}+(1-\alpha)\ell_T =\alpha\ell_T+(1-\alpha)\ell_T =\ell_T.

The next forecast is therefore $\hat{y}_{T+2|T}=\hat{\ell}_{T+1|T}=\ell_T$. Repeating the same reasoning gives the result for every horizon. More formally, suppose $\hat{\ell}_{T+h-1|T}=\ell_T$ for some $h\geq 1$. The forecasted observation at that horizon is $\hat{y}_{T+h|T}=\hat{\ell}_{T+h-1|T}=\ell_T$. Consequently,

\hat{\ell}_{T+h|T} =\alpha\hat{y}_{T+h|T}+(1-\alpha)\hat{\ell}_{T+h-1|T} =\alpha\ell_T+(1-\alpha)\ell_T =\ell_T.

The base case and recursive step establish that the forecasted level remains $\ell_T$, so $\hat{y}_{T+h|T}=\ell_T$ for all $h\geq1$. This recursion is a forecasting calculation, not a claim that the actual future level is known. When the actual $y_{T+1}$ is observed, the update uses that value instead of its forecast.

Calculate the current level.
Use the specified initialization and apply the smoothing update through the last observed time $T$.

Set the first forecast.
The one-step forecast from origin $T$ is $\ell_T$.

Extend to the requested horizons.
For every $h\geq1$, report $\hat{y}_{T+h|T}=\ell_T$. Do not apply the smoothing update to an invented future observation.

Worked Calculations

Worked Example: Forecasts After Updating the Level

A hypothetical portfolio administrator tracks a monthly index of claim-processing workload. Suppose the initialization is $\ell_0=100$, the smoothing parameter is $\alpha=0.30$, and the observations are $y_1=110$ and $y_2=104$. Calculate the level after each observation, then give the forecasts from time $T=2$ for horizons one through four. Use SES with the stated initialization and parameter.

Method and level at time 1. Apply $\ell_1=\alpha y_1+(1-\alpha)\ell_0$:

\ell_1=0.30(110)+0.70(100)=33+70=103.

Level at time 2. Use the newly observed value $y_2=104$ and the previous level $\ell_1=103$:

\ell_2=0.30(104)+0.70(103)=31.2+72.1=103.3.

Forecasts from time 2. The current level is $\ell_2=103.3$. Therefore,

\hat{y}_{3|2}=\hat{y}_{4|2}=\hat{y}_{5|2}=\hat{y}_{6|2}=103.3.

Check. Each forecast uses the same origin, time 2, and the same last calculated level, 103.3. The common mistake avoided here is continuing to smooth using the past observations again when calculating horizons three and beyond.

Worked Example: Forecasting From a Different Initialization

A hypothetical insurer monitors a weekly service-volume index. Let the initial level be $\ell_0=80$, let $\alpha=0.25$, and suppose the first three observations are $92$, $88$, and $96$. Calculate the final level and the forecasts at horizons one, two, and five from $T=3$. Use the SES level recursion; no future observations are available at the forecast origin.

First update. For $y_1=92$,

\ell_1=0.25(92)+0.75(80)=23+60=83.

Second update. For $y_2=88$,

\ell_2=0.25(88)+0.75(83)=22+62.25=84.25.

Third update. For $y_3=96$,

\ell_3=0.25(96)+0.75(84.25)=24+63.1875=87.1875.

Forecasts from time 3. The requested horizons all use the same level:

\hat{y}_{4|3}=87.1875,\qquad \hat{y}_{5|3}=87.1875,\qquad \hat{y}_{8|3}=87.1875.

Check. The observations have moved the level from 80 to 87.1875, but the forecast does not change across horizons from the fixed origin $T=3$. The common mistake avoided is treating the horizon number as a multiplier of the smoothing parameter.

Worked Example: Revising Forecasts After a New Observation

At time $T$, a hypothetical claims operation has SES level $\ell_T=50$ and smoothing parameter $\alpha=0.40$. First state the three-step forecast from time $T$. Then suppose the next observed value is $y_{T+1}=56$. Calculate the updated level and the two-step forecast from the new origin $T+1$. Use the observed value only for the update after it becomes available.

Forecasts from time $T$. Before $y_{T+1}$ is observed, every horizon forecast equals 50. In particular, $\hat{y}_{T+3|T}=50$.

Calculate the new level. The observation differs from the previous level by $56-50=6$. Applying the update gives

\ell_{T+1}=0.40(56)+0.60(50)=22.4+30=52.4.

Forecast from the revised origin. With time $T+1$ now the origin, the two-step forecast is

\hat{y}_{T+3|T+1}=\ell_{T+1}=52.4.

Check. The forecast for time $T+3$ was 50 when made at $T$, and is 52.4 when recalculated at $T+1$. These are forecasts from different origins, not inconsistent answers. The common mistake avoided is using $y_{T+1}$ in the forecast made at $T$, when that observation was not yet available.

Interpretation and Common Traps

A flat sequence is the defining multi-step point-forecast behavior of SES. It reflects the structure of the method: the level is the only component used to project the series forward. It should not be interpreted as evidence that actual future values will be constant. Actual observations may lie above or below the forecast; once observed, they affect the updated level through $\alpha$.

For example, if $\ell_T=103.3$ and $\alpha=0.30$, one incorrect approach is to calculate the two-step forecast as $\alpha\ell_T=30.99$, as if the smoothing parameter were a multiplier that reduces the future forecast. The correct forecasted level update uses both the forecasted observation and the previous forecasted level:

$$ 0.30(103.3)+0.70(103.3)=30.99+72.31=103.3. $$

Thus the two-step forecast remains 103.3. Another error is to add the most recent observed change to every future forecast. That imposes a trend extrapolation not present in simple exponential smoothing. Here the forecast must be derived from the SES rule, not from an unmodeled change between the last two observations.

A second distinction is between a point forecast and certainty about the future. The equality $\hat{y}_{T+h|T}=\ell_T$ gives a forecasted value, not a guarantee that $y_{T+h}=\ell_T$. As the horizon extends, more future innovations can affect the realized observations. The flat point forecast alone does not describe the full range of possible outcomes.

Key takeaway: First compute the SES level using all observations available at the forecast origin. Then use that single level for every future point forecast from that origin: $\hat{y}_{T+h|T}=\ell_T$, $h\geq1$. Recalculate the level and forecasts only when new data are observed.

Check Your Understanding

For each question, use the stated initialization, smoothing parameter, and SES update. Distinguish forecasts made from a fixed origin from forecasts recalculated after a new observation.

At time $T$, a hypothetical workload series has level $\ell_T=125$. What are the forecasts at horizons one, three, and eight? (Answer: 125, 125, and 125, respectively.)
Let $\ell_0=40$, $\alpha=0.50$, and observations $y_1=46$, $y_2=42$. Calculate $\ell_1$, $\ell_2$, and the forecasts at horizons one and four from time 2. (Answer: $\ell_1=43$, $\ell_2=42.5$; both forecasts are 42.5.)
At time $T$, the level is 70 and $\alpha=0.30$. If the next observation is $y_{T+1}=64$, calculate the updated level and the two-step forecast from time $T+1$. (Answer: updated level 68.2; forecast 68.2.)
A hypothetical index has level 100 at time $T$ and $\alpha=0.40$. What is the four-step forecast made at $T$? If $y_{T+1}=110$, what is the three-step forecast made at $T+1$? (Answer: 100 from time $T$; 104 from time $T+1$.)
Let $\ell_0=200$, $\alpha=0.20$, and $y_1=180$, $y_2=190$. Find $\ell_1$, $\ell_2$, and the five-step forecast from time 2. If $y_3=210$, find the two-step forecast from time 3. (Answer: $\ell_1=196$, $\ell_2=194.8$, five-step forecast from time 2 is 194.8, and two-step forecast from time 3 is 197.84.)

What You'll Learn