\section{The Automobile Market Liquidity Index}\label{sec:amli}

\subsection{Construction}

Table~\ref{tab:amli} summarises the index. Five components are computed for each of the seventeen regions and each panel week from week~2 to week~10 (the two first weeks are contaminated by the portal start-ups): exit intensity, markdown frequency, markdown depth, quality-adjusted dispersion (sign reversed) and arrival rate. Each is standardised across the 153 region-weeks; higher values mean a more liquid market under the preregistered sign convention. Figure~\ref{fig:amli} plots the province-level components and index and the regional indices with dealer-bootstrap bands.

\begin{table}[tbp]\centering\caption{AMLI construction, weights and internal validation (region $\times$ week, weeks 2--10)}\label{tab:amli}
\tabsize\input{tables/tab07_amli}
\begin{minipage}{\textwidth}\scriptsize\smallskip Notes: PCA loading = first principal component of the five standardised components, signed so that exit intensity loads positively. Supervised = ridge coefficients of next-week exit intensity on the four non-exit components, fitted on weeks 2--5. Leave-one-out corr. = correlation between the equal-weight index without the component and the full index. Held-out validation: coefficient of next-week exit intensity on an equal-weight index built without the exit component, region fixed effects, weeks 6--10, standard errors clustered by region; slope in percentage points of daily exit intensity per index unit.\end{minipage}
\end{table}

\begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{figures/fig08_amli_series.pdf}
\caption{AMLI time series. (a) Province-level standardised components (signed). (b) Province-level index under three weighting schemes. (c) Equal-weight index for the six largest regions with 95\% dealer-block-bootstrap bands (200 replications).}\label{fig:amli}\end{figure}

\subsection{What the components say about liquidity}

The component correlations (Figure~\ref{fig:amlival}(b)) carry the paper's most useful lesson about liquidity measurement. Exit intensity co-moves positively with the arrival rate ($\rho = 0.62$) and, weakly, with markdown frequency (0.19), but \emph{negatively} with markdown depth ($-0.35$) and positively with reversed dispersion (0.44): region-weeks in which vehicles leave quickly are region-weeks with many new arrivals, frequent but \emph{shallow} price cuts, and \emph{tight} quality-adjusted price distributions; region-weeks with deep cuts and dispersed prices are slow. Markdown depth and dispersion are themselves strongly correlated (0.85). The first principal component explains 56~percent of the variance of the five components, above the 40~percent threshold set in the analysis plan, and its loadings are positive on exit, markdown frequency, arrival and reversed dispersion, but \emph{negative} on markdown depth. The preregistered sign for depth (deeper cuts = more liquid) is therefore contradicted by the data, which treat deep cuts as a symptom of illiquidity, in the spirit of the list-price-cut literature in housing \citep{knight2002listing,anenberg2016frictions}. We keep the preregistered equal-weight index as the primary measure, report the PCA-weighted index (correlation 0.85 with the equal-weight index) and the supervised index (0.68) alongside it, and record this sign reversal as a substantive finding rather than a tuning choice. At the province level, with only nine weekly observations, the first component explains 44~percent of the variance and its loadings are unstable; the regional panel is the appropriate level for the index in a window this short.

\subsection{Internal validation}

The analysis plan called for a held-out-component test. An equal-weight index built \emph{without} exit intensity, from the markdown, dispersion and arrival components of week $w$, predicts the exit intensity of week $w+1$ within regions: the slope is 0.97 percentage points of daily exit intensity per index unit (95\% interval 0.69 to 1.24; the mean daily exit intensity is 2.3~percent), with an $R^2$ of 0.66 including region effects, and it is unchanged when the current exit intensity is added as a control (1.03, interval 0.70 to 1.35). Figure~\ref{fig:amlival}(a) shows the within-region relation. The index is robust to leaving any single component out (correlations with the full index between 0.91 and 0.95; the held-out slope stays between 0.58 and 0.82 points), and to the minimum cell size (100 listing-days: unchanged), but the held-out validation is weaker under rank standardisation of the components (0.46 points, standard error 0.29) and uninformative at two-week aggregation, which leaves only four observations per region (Section~\ref{sec:robust}). The same construction on price-tier and body-class markets gives first components explaining 44--46~percent of the variance and held-out slopes of 0.46 points, significant for body classes (interval 0.06 to 0.86) but not for the three price tiers (12 market-weeks only).

\begin{figure}[tbp]\centering\includegraphics[width=\textwidth]{figures/fig09_amli_validation.pdf}
\caption{AMLI internal validation. (a) Within-region relation between the leave-exit-out index in week $w$ and exit intensity in week $w+1$, weeks 6--10. (b) Correlations among the five standardised components across region-weeks.}\label{fig:amlival}\end{figure}

\subsection{What the index cannot do yet}

The window contains eleven weeks. No weekly or daily independent benchmark of used-vehicle turnover exists for Quebec, and the monthly or quarterly series that do exist (Statistics Canada new-vehicle sales, the Canadian Black Book retention index, the Statistics Canada used-vehicle CPI, AutoTrader's quarterly price index) offer at most three overlapping observations, most of them national. We therefore make no claim that the AMLI forecasts any economic variable, and we regard its validation as internal and partial: it summarises coherent variation in observed turnover across regions and weeks, and its non-exit components anticipate next week's exits. Establishing external validity requires a panel of at least one to two years and, ideally, access to registration transfers, which is the natural next step for the platform.