Practice with worked answers

The March record is unavailable on April 30 because May 10 follows the origin. Its March period label does not override the release date. A real-time reconstruction requires the record available at April 30; a latest-vintage retrospective analysis should be labeled accordingly.

Weighted temperature is $0.75(30)+0.25(90)=45$. Transforming it gives $65-45=20$. Local transformations give 35 and zero, so their weighted value is $0.75(35)=26.25$. The positive-part transformation is convex; aggregating before transforming loses exposure variation.

Sum monthly spending flows to obtain quarterly spending. For debt, use a defined quarter-end stock or an average stock if that is the question. Summing three month-end stocks double-counts continuing debt positions and changes the quantity’s interpretation.

Mean is $2/(1-0.75)=8$. The forecast is $8+0.75^2(12-8)=10.25$. Error variance is $4(1+0.75^2)=6.25$, giving standard deviation 2.5. This excludes estimation uncertainty and relies on the specified innovation assumptions.

The response alternates in sign and halves in magnitude each period. After a unit shock, the sequence is

$1,-0.5,0.25,-0.125,\ldots.$

Variance remains nonnegative because it uses squared response coefficients.

Quarterly averages combine three correlated monthly states. Their adjacent-quarter covariance includes nine pairwise distances, not only a three-period lag. Sampling the quarter’s last month is different from averaging the quarter. Use the covariance sums in Lecture 2 to calculate the appropriate aggregate correlation.

Variance is $16(1+0.25^2)=17$; lag-one covariance is $16(0.25)=4$; correlation is $4/17$. Later positive-lag covariances are zero under the white-noise assumptions.

The reciprocal coefficient is 4 and innovation variance is 1. Its variance is $1(1+16)=17$ and lag-one covariance is 4. The coefficient 0.25 representation is invertible; the coefficient 4 representation is not. Equality of covariance functions supplies a second-order equivalence, with stronger distributional equivalence under appropriate Gaussian assumptions.

The AR and MA lag polynomials cancel, leaving white noise apart from initialization conventions. The redundant parameters can appear to offer flexibility without identifying separate mechanisms. Inspect common factors and simplify the representation.

The regressor is inadmissible at $t$ because both its described period and release follow the origin. A separately forecast or scenario value could be used with a clear label and appropriate uncertainty treatment.

The optimum satisfies $F(a)=3/(3+1)=0.75$. A conditional-mean forecast is not automatically the cost-minimizing action under this asymmetric loss.

Paired differences are $(-3,5,-5)$, averaging $-1$. The first method has lower mean squared loss on these three targets. The second target reverses the ranking, and three observations provide weak evidence of general performance. Dependence and target representativeness also matter.

The latent variance is $0.64/(1-0.36)=1$, and its lag-one covariance is 0.6. Observed variance is $1+3=4$, so observed correlation is 0.15. The lower correlation need not imply low latent persistence.

The centered value increases by $(30+30)/5=12$. A fixed-parameter trailing filter using only dates through $t$ would be unchanged. A full-sample parameter refit can introduce another source of future dependence and must be audited separately.

The trailing mean averages states at $t,t-1,t-2$, so its expected level is the state at $t-1$. It is three units below the current state. Noise reduction and timely state estimation are different objectives.

Forecast is $40+5=45$ and variance is $5(2)=10$. A shock permanently affects the level; a fitted deterministic trend does not remove the accumulated innovation path.

Differencing gives an MA(1) with coefficient $-1$. Variance is 4 and lag-one covariance is $-2$, so correlation is $-0.5$. The added dependence is a reason to investigate over-differencing.

The statistic does not cross the stated rejection threshold. This is non-rejection under that specification, not proof of a unit root. Lag choice, deterministic terms, breaks, and near-unit alternatives affect interpretation. The usual normal cutoff is not the DF reference.

Transformed outcome is $14-0.4(9)=10.4$; predictor is $5-0.4(2)=4.2$; intercept is $5(1-0.4)=3$. The transformed regression component is $3+2(4.2)=11.4$, leaving innovation $-1$. Transforming only the outcome and predictor while leaving the intercept unchanged misparameterizes the equation.

The forecast is $20+0.4(2)=20.8$. It conditions on the supplied regression inputs. If those inputs include uncertain future weather, their forecast or scenario construction and uncertainty must be stated.

A calendar month is known before it occurs. Actual weather is not. A climate expectation, weather forecast, or scenario can be admissible if constructed from information available at the origin.

Gain is $3/(3+6)=1/3$. Innovation is 6, so posterior mean is 10 and posterior variance is $(1-1/3)3=2$. Mean units and squared variance units remain distinct.

After the prediction step, retain prior mean 8 and variance 3. Inserting zero would treat absence as a real zero-valued measurement and pull the estimate toward it. Missingness may itself require a more complex model if it is informative.

The smoother conditions on observations after the historical date. Those observations can refine a past-state estimate but were unavailable at the original decision. Filtered and smoothed outputs answer different information-set questions.

One-step forecast is $(5.5,5)'$. Multiplying again gives $(3.7,4.1)'$. In this column-vector convention, a row of $A$ maps source histories into one destination’s prediction.

The matrix’s eigenvalues are 1.1 and 0.3, so the system is unstable despite both diagonal entries being below one. Stability concerns the joint feedback system.

Variance of the sum is $4+9+2(1)=15$. Ignoring contemporaneous covariance understates risk here. A known network and a covariance matrix supply different information; neither by itself identifies an intervention.

The factor is $L=\begin{pmatrix}1&0\\0.6&1\end{pmatrix}$. Its product with its transpose gives the stated covariance. The zero imposes a recursive immediate exclusion.

With $Q=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, $LQ=\begin{pmatrix}0&-1\\1&-0.6\end{pmatrix}$. Covariance is unchanged because $QQ'=I$. The first shock’s impact pattern changes, so covariance has not uniquely identified its economic meaning.

A sampling band describes estimation variation under maintained identification and other assumptions. Its narrowness does not establish that the ordering is economically correct. Examine credible alternative restrictions and the claim’s sensitivity.

The spread is $100-1.5(60)=10$. Predicted change is $-0.2(10)+1.5(2)=1$, so the next predicted level is 101. The equilibrium relation, adjustment coefficient, and input forecast each have a separate role.

In the simple constructed spread, persistence is $1+\alpha=-2$, so $\alpha=-3$ implies explosive alternating deviations. Stability requires $-2<\alpha<0$ in that model. A general multivariate system requires its own joint stability check.

The residual incorporates estimated long-run parameters, altering the null distribution. An ordinary observed-series DF threshold does not automatically apply. The specification’s deterministic terms and dimensions must also be retained.

Next variance is $0.1+0.2(4)+0.7(1.5)=1.95$. Long-run variance is $0.1/(1-0.9)=1$. The following variance forecast is $1+0.9(1.95-1)=1.855$.

Divide residuals by 100 and variances and $\omega$ by 10,000. Thus next variance is 0.000195 and $\omega$ is 0.00001; $\alpha$ and $\beta$ remain unchanged. Quantiles and standard deviations divide by 100.

The quantile also needs a conditional mean and standardized-shock distribution. Under zero-mean Gaussian shocks, it is approximately $-1.645\sqrt{1.95}$ percentage points. A heavy-tailed distribution can produce a different threshold with the same conditional variance.

Differences are $(3,-3,5)$, averaging $5/3$. The rolling procedure is worse on average over these targets, even though it wins on the second. Window choice should not be based on a generic claim that recency is always better.

A shorter window can reduce contamination from an obsolete regime after a break. In a stable process it discards relevant information and can increase estimation variation. Measurement changes can also resemble breaks and require a different repair.

Under the simple stated cost accounting, replacement loses four units per month. A justification would require additional benefits, a different risk objective, or evidence that the cost estimates are incomplete. A small RMSE improvement alone does not establish operational value.

Validation selects the second method, whose continuation loss is 1.6. Reporting 0.9 would substitute a method selected using continuation outcomes. That is a retrospective selection result, not the original rule’s result.

Yes, if the refitting schedule and admitted information were part of the frozen algorithm. Freezing the procedure is different from fixing every fitted parameter. Changing the window based on continuation errors is a new selection.

Use the declared fallback or abstention rule and retain a failure flag. Dropping the row changes the target set and can inflate performance. Operational reliability is part of the evaluated procedure.

The forecast should be 1,200 and variance 90,000. Scale means by 100 and squared quantities by 10,000. If a consistent implementation violates this relationship, investigate transformations and units.

Predictive improvement does not identify an intervention effect. The report needs a credible identification strategy with assumptions and supporting evidence, or it must narrow the causal claim while retaining the useful predictive result.

Define the target first, verify admissible information, then audit computation and interpret uncertainty under the correct model. An exact interval for a misdefined quantity or infeasible forecast does not answer the intended decision.

The target is next month’s residential sales in a specified region, measured in million MWh. At an end-of-month origin, the assumed sales lag makes the last admitted observation two months older than the origin. A recursion using that latest level must forecast three periods to reach the next-month target. The centered activity index is unavailable at its historical date because it includes two later observations. Replace it with an admissible construction or explicitly treat it as retrospective evidence; then reevaluate the affected procedure.

The selected regression’s window and update rule were frozen using the repair period, so its later refits can be admissible. The twelve continuation outcomes evaluate that retained rule. Reselecting using those outcomes would create a new exploratory procedure. The benchmark has slightly lower realized RMSE, and the small sample does not establish durable superiority. Retaining the benchmark as the main rule is defensible while evaluating a repaired regression on a new boundary.

The latest downloaded vintage does not reconstruct every historical release. The comparison is pseudo-out-of-sample unless historical vintages are supplied. A Gaussian interval assumes the specified conditional distribution and variance treatment; a simple plug-in interval may omit parameter, predictor, selection, and measurement uncertainty. Coverage 8/12 is about 66.7%, which warrants investigation, but a short and possibly dependent target sequence limits formal conclusions.

Cholesky factorization reproduces residual covariance under an ordering restriction. It does not establish that the restriction correctly identifies a regional intervention. State the recursive immediate exclusions, examine credible alternatives, and narrow the causal claim unless additional identifying evidence supports it.

A repair should address the actual issue: remove future information from the index, improve the justified uncertainty calculation, and preserve the benchmark and target definition. Freeze the revised algorithm before evaluating another continuation segment. Reconsider the recommendation if later matched losses show economically meaningful and reliable improvement, if the benchmark becomes infeasible, or if the target’s measurement changes.

For the 4 reasoning points, full-credit evidence correctly defines the clock, diagnoses the centered input, preserves selection boundaries, and recommends an evidence-supported action. For the 3 assumption points, full-credit evidence explains vintages, interval scope, and identification. For the 3 interpretation points, it interprets the actual comparison, proposes a targeted repair, and states reconsideration evidence. A minor arithmetic slip need not erase otherwise coherent reasoning; a consequential information or identification error affects the criterion it undermines.