Last-Value Naive Forecasts

Introduction

A simple forecast can be useful precisely because it makes few assumptions. If a time series has just been observed at a particular level, a natural starting point is to predict that it will remain at that level for the next period. This is the last-value naive forecast: the most recent observation is carried forward unchanged.

The method is easy to calculate, but it is also easy to use incorrectly. The forecast must be based only on observations available at its origin, and its error must be calculated with a consistent sign convention. These details matter when a series has a sudden level change: the forecast does not anticipate the change, so the resulting error can be substantial.

Definition: Let $y_1,\ldots,y_T$ be observed values of a time series through time $T$. The last-value naive forecast made at time $T$, for any future horizon $h\geq 1$, is $$ \hat{y}_{T+h\mid T}=y_T. $$ For one-step forecasting through the observed series, the forecast for $y_t$, made using data only through $t-1$, is $\hat{y}_{t\mid t-1}=y_{t-1}$. We define the signed forecast error as actual minus forecast: $e_t=y_t-\hat{y}_{t\mid t-1}$.

The subscript $T+h\mid T$ identifies both the target time and the forecast origin: it is a forecast for time $T+h$, made with information available at time $T$. For example, if the latest observation is $y_6=40$, then forecasts made at time 6 are 40 for time 7, 40 for time 8, and 40 for every later horizon. New observations can change the forecast: after observing $y_7$, the forecast made at time 7 uses $y_7$, not $y_6$.

How the Forecast Is Constructed

The procedure requires no fitted trend or seasonal pattern. At a forecast origin, identify the latest available observation and copy its value to each requested future period. If actual future values are later supplied, subtract each corresponding forecast from its actual value to calculate the signed error. A positive error means the actual value exceeded the forecast; a negative error means the actual value was below it.

Set the origin.
Write down the last time point whose observation is available. Do not use observations that occur after this origin to construct its forecasts.

Carry the latest value forward.
For every requested horizon $h\geq 1$, set the forecast equal to the observation at the origin.

Calculate errors when actuals are known.
For each target period, calculate actual minus its forecast. Keep the time index attached to each result.

A sequence that changes level highlights the method's limitation. If a series moves from values near 80 to values near 100, a forecast made before that shift still carries forward the older observation. A forecast made after the first new observation uses the changed level instead. Thus, a forecast error records the gap at a particular target time; it does not by itself tell us why the series changed.

Why This Method Has a Random-Walk Rationale

The last-value method can be motivated by a random-walk model. Suppose the series follows $y_t=y_{t-1}+\varepsilon_t$, where the innovations $\varepsilon_t$ are independent over time, have mean zero and variance $\sigma^2$, and are independent of information available through time $t-1$. No particular innovation distribution, such as a normal distribution, is required for the following conditional-mean result.

Result: Under the stated random-walk assumptions, the conditional expectation of the future value, given observations through time $T$, is the last observed value: $$ E[y_{T+h}\mid y_1,\ldots,y_T]=y_T,\qquad h\geq 1. $$

To see why, repeatedly substitute the model equation:

y_{T+h}=y_T+\varepsilon_{T+1}+\varepsilon_{T+2}+\cdots+\varepsilon_{T+h}.

Taking conditional expectations given the information through $T$, the known value $y_T$ remains unchanged. Each future innovation has conditional expectation zero by the assumptions. Therefore,

E[y_{T+h}\mid y_1,\ldots,y_T] =y_T+\sum_{j=1}^{h}E[\varepsilon_{T+j}\mid y_1,\ldots,y_T] =y_T.

This derivation explains when carrying the latest value forward is a conditional-mean forecast: the random-walk assumptions make future changes have zero conditional mean. The last-value procedure can still be applied as a simple benchmark when those assumptions are not established, but it should not then be described as a proven optimal forecast. In particular, a series with systematic trend or seasonality may require a method that represents those features.

Worked Example: One-Step Forecasts Across a Level Change

A hypothetical insurer tracks a monthly service-volume index. The recorded values for five consecutive months are 82, 85, 84, 101, and 98. Construct a last-value one-step forecast for each month after the first, and calculate the signed forecast errors. Use actual minus forecast.

Method. Each one-step forecast uses the value from the immediately preceding month. The observations are $y_1=82$, $y_2=85$, $y_3=84$, $y_4=101$, and $y_5=98$.

For month 2, the forecast is $y_1=82$, so the error is $85-82=3$. For month 3, the forecast is $y_2=85$, so the error is $84-85=-1$. For month 4, the forecast is $y_3=84$, so the error is $101-84=17$. For month 5, the forecast is $y_4=101$, so the error is $98-101=-3$.

Target month	Actual	Forecast from prior month	Error: actual minus forecast
2	85	82	3
3	84	85	-1
4	101	84	17
5	98	101	-3

The error of 17 in month 4 reflects the sharp increase from 84 to 101: a last-value forecast cannot anticipate a change before observing it. The method has updated by month 5, so its forecast is 101. Check: each forecast equals the previous month's actual, and each displayed error equals actual minus forecast. The common mistake avoided here is forecasting month 4 with 101, which uses the target month’s actual value and therefore is not a genuine forecast.

Worked Example: Several Horizons From One Origin

A hypothetical quarterly claims-processing index has most recently been observed at 126 in quarter 4. A forecast is made at the end of quarter 4 for quarters 5 through 7. The subsequently observed values are 131, 124, and 139. Find each forecast and its signed error.

Method. The forecast origin is quarter 4, and the latest value available then is $y_4=126$. The forecast is therefore 126 at every requested horizon:

\hat{y}_{5\mid4}=126,\qquad \hat{y}_{6\mid4}=126,\qquad \hat{y}_{7\mid4}=126.

For quarter 5, the error is $131-126=5$. For quarter 6, it is $124-126=-2$. For quarter 7, it is $139-126=13$.

Target quarter	Horizon from quarter 4	Actual	Forecast	Error
5	1	131	126	5
6	2	124	126	-2
7	3	139	126	13

The three forecasts remain fixed because all were made at the same origin. If a new forecast were made after quarter 5 was observed, its latest value would instead be 131. Check: the errors follow the stated convention: $131-126=5$, $124-126=-2$, and $139-126=13$. The common mistake avoided is updating the later forecasts with intervening actuals when the question asks for forecasts all made at quarter 4.

Worked Example: Updating After Each New Observation

A hypothetical weekly count of completed applications is $42,42,39,47,47$. Calculate the sequential one-step forecasts and errors for weeks 2 through 5. Then, using the observation at week 5, forecast weeks 6 and 7.

Method. For each observed week after the first, use the preceding week's count. For week 2, the forecast is 42 and the error is $42-42=0$. For week 3, the forecast is 42 and the error is $39-42=-3$. For week 4, the forecast is 39 and the error is $47-39=8$. For week 5, the forecast is 47 and the error is $47-47=0$.

At the end of week 5, the latest observation is 47. The forecasts made then are $\hat{y}_{6\mid5}=47$ and $\hat{y}_{7\mid5}=47$. Those are forecasts only; no errors for weeks 6 and 7 can be calculated unless their actual observations are provided.

Check: the sequential errors are $0,-3,8,0$, and both future forecasts equal the latest observed value, 47. The common mistake avoided is treating a forecast as though it were an actual observation and calculating an error before an actual value exists.

Interpreting Errors and Avoiding Traps

Forecast errors are useful here as a direct check on what the method missed. They should be matched to the forecast origin and horizon that produced them. Under the convention used throughout this tutorial, $e_t=y_t-\hat{y}_{t\mid t-1}$. If the actual exceeds the forecast, the error is positive; if the actual falls short, it is negative. Reversing the subtraction reverses every sign and changes the stated convention.

The principal limitation is the method's response to a level change. Before a jump, the forecast remains at the old level. Once the new value is observed, subsequent forecasts carry the new value forward. The method does not smooth a jump, infer its cause, or extrapolate it into a trend. It is a transparent benchmark, and its simplicity makes it valuable for checking whether a more elaborate forecast is improving on a straightforward reference.

Do not use future information. A forecast made at time $T$ must be based on observations through $T$, not on $y_{T+1}$ or later values.
Do not confuse forecast origin with forecast target. Multiple horizons from the same origin all use the same latest value.
State the error convention. Here, error is actual minus forecast; a different convention must be stated explicitly if used.
Do not calculate an error without an actual. A future forecast alone is not enough to determine a forecast error.

Key takeaway: A last-value naive forecast carries the latest observation forward unchanged at every future horizon. For an observed target, its signed error is the target actual minus the forecast made for that target. The forecast's quality can deteriorate sharply when the series changes level, especially before the new level has been observed.

Check Your Understanding

For each problem, use the last-value method and the signed-error convention actual minus forecast.

A hypothetical monthly exposure index is 18, 21, 20, and 26. Find the one-step forecasts and errors for months 2 through 4. (Answer: Forecasts $18,21,20$; errors $3,-1,6$.)
A quarterly service count is 310, 304, 318, 315, and 330. At the end of quarter 5, actuals for quarters 6 through 8 are later observed as 326, 335, and 329. Give the quarter-5-origin forecasts and errors. (Answer: Forecasts $330,330,330$; errors $-4,5,-1$.)
A sequence of weekly values is 7, 9, 8, 8, and 13. Calculate the sequential one-step forecast errors for weeks 2 through 5. (Answer: $2,-1,0,5$.)
At forecast origin 3, the latest observed value is 56. Actual values at horizons 1, 2, and 3 are 53, 61, and 60. Give the three forecasts and signed errors. (Answer: Forecasts $56,56,56$; errors $-3,5,4$.)
A hypothetical index has observations 100, 105, and 99. At the end of the third period, forecast period 4; its actual value is 102. What is the forecast and its error? (Answer: Forecast $99$; error $3$.)

What You'll Learn